Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 84 items • Updated • 3
Datasets:
fact stringlengths 9 24k | type stringclasses 2
values | library stringclasses 5
values | imports listlengths 0 0 | filename stringclasses 5
values | symbolic_name stringlengths 1 24 | docstring stringlengths 12 292k ⌀ |
|---|---|---|---|---|---|---|
idi : |- ph | theorem | set | [] | set.mm | idi | This is the Metamath database set.mm. $) $( Metamath is a formal language and associated computer program for archiving, verifying, and studying mathematical proofs, created by Norman Dwight Megill (1950--2021). For more information, visit https://us.metamath.org and https://github.com/metamath/set.mm, and feel free to... |
a1ii : |- ph | theorem | set | [] | set.mm | a1ii | (_Note_: This inference rule and the previous one, ~ idi , will normally never appear in a completed proof.) This is a technical inference to assist proof development. It provides a temporary way to add an independent subproof to a proof under development, for later assignment to a normal proof step. The Metamath (Meta... |
mp2 : |- ch | theorem | set | [] | set.mm | mp2 | Register implication '->' as a primitive expression (lacking a definition). $) $( $j primitive 'wi'; $) $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= The axioms of propositional calculus =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Propositional ca... |
mp2b : |- ch | theorem | set | [] | set.mm | mp2b | A double modus ponens inference. (Contributed by Mario Carneiro, 24-Jan-2013.) |
a1i : |- ( ps -> ph ) | theorem | set | [] | set.mm | a1i | null |
2a1i : |- ( ps -> ( ch -> ph ) ) | theorem | set | [] | set.mm | 2a1i | Inference introducing two antecedents. Two applications of ~ a1i . Inference associated with ~ 2a1 . (Contributed by Jeff Hankins, 4-Aug-2009.) |
mp1i : |- ( ch -> ps ) | theorem | set | [] | set.mm | mp1i | Inference detaching an antecedent and introducing a new one. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
a2i : |- ( ( ph -> ps ) -> ( ph -> ch ) ) | theorem | set | [] | set.mm | a2i | Inference distributing an antecedent. Inference associated with ~ ax-2 . Its associated inference is ~ mpd . (Contributed by NM, 29-Dec-1992.) |
mpd : |- ( ph -> ch ) | theorem | set | [] | set.mm | mpd | A modus ponens deduction. A translation of natural deduction rule ` -> ` E ( ` -> ` elimination), see ~ natded . Deduction form of ~ ax-mp . Inference associated with ~ a2i . Commuted form of ~ mpcom . (Contributed by NM, 29-Dec-1992.) |
imim2i : |- ( ( ch -> ph ) -> ( ch -> ps ) ) | theorem | set | [] | set.mm | imim2i | null |
syl : |- ( ph -> ch ) | theorem | set | [] | set.mm | syl | null |
3syl : |- ( ph -> th ) | theorem | set | [] | set.mm | 3syl | Inference chaining two syllogisms ~ syl . Inference associated with ~ imim12i . (Contributed by NM, 28-Dec-1992.) |
4syl : |- ( ph -> ta ) | theorem | set | [] | set.mm | 4syl | Inference chaining three syllogisms ~ syl . (Contributed by BJ, 14-Jul-2018.) The use of this theorem is marked "discouraged" because it can cause the Metamath program "MM-PA> MINIMIZE__WITH *" command to have very long run times. However, feel free to use "MM-PA> MINIMIZE__WITH 4syl / OVERRIDE" if you wish. Remember t... |
mpi : |- ( ph -> ch ) | theorem | set | [] | set.mm | mpi | null |
mpisyl : |- ( ph -> th ) | theorem | set | [] | set.mm | mpisyl | A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.) |
id : |- ( ph -> ph ) | theorem | set | [] | set.mm | id | null |
idALT : |- ( ph -> ph ) | theorem | set | [] | set.mm | idALT | null |
idd : |- ( ph -> ( ps -> ps ) ) | theorem | set | [] | set.mm | idd | Principle of identity ~ id with antecedent. (Contributed by NM, 26-Nov-1995.) |
a1d : |- ( ph -> ( ch -> ps ) ) | theorem | set | [] | set.mm | a1d | null |
2a1d : |- ( ph -> ( ch -> ( th -> ps ) ) ) | theorem | set | [] | set.mm | 2a1d | Deduction introducing two antecedents. Two applications of ~ a1d . Deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by BJ, 10-Aug-2020.) |
a1i13 : |- ( ph -> ( ps -> ( ch -> th ) ) ) | theorem | set | [] | set.mm | a1i13 | Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.) |
2a1 : |- ( ph -> ( ps -> ( ch -> ph ) ) ) | theorem | set | [] | set.mm | 2a1 | A double form of ~ ax-1 . Its associated inference is ~ 2a1i . Its associated deduction is ~ 2a1d . (Contributed by BJ, 10-Aug-2020.) (Proof shortened by Wolf Lammen, 1-Sep-2020.) |
a2d : |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) | theorem | set | [] | set.mm | a2d | Deduction distributing an embedded antecedent. Deduction form of ~ ax-2 . (Contributed by NM, 23-Jun-1994.) |
sylcom : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | sylcom | Syllogism inference with commutation of antecedents. (Contributed by NM, 29-Aug-2004.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) (Proof shortened by Stefan Allan, 23-Feb-2006.) |
syl5com : |- ( ph -> ( ch -> th ) ) | theorem | set | [] | set.mm | syl5com | Syllogism inference with commuted antecedents. (Contributed by NM, 24-May-2005.) |
com12 : |- ( ps -> ( ph -> ch ) ) | theorem | set | [] | set.mm | com12 | null |
syl11 : |- ( ps -> ( th -> ch ) ) | theorem | set | [] | set.mm | syl11 | A syllogism inference. Commuted form of an instance of ~ syl . (Contributed by BJ, 25-Oct-2021.) |
syl5 : |- ( ch -> ( ph -> th ) ) | theorem | set | [] | set.mm | syl5 | A syllogism rule of inference. The first premise is used to replace the second antecedent of the second premise. (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 25-May-2013.) |
syl6 : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | syl6 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 5-Jan-1993.) (Proof shortened by Wolf Lammen, 30-Jul-2012.) |
syl56 : |- ( ch -> ( ph -> ta ) ) | theorem | set | [] | set.mm | syl56 | Combine ~ syl5 and ~ syl6 . (Contributed by NM, 14-Nov-2013.) |
syl6com : |- ( ps -> ( ph -> th ) ) | theorem | set | [] | set.mm | syl6com | Syllogism inference with commuted antecedents. (Contributed by NM, 25-May-2005.) |
mpcom : |- ( ps -> ch ) | theorem | set | [] | set.mm | mpcom | Modus ponens inference with commutation of antecedents. Commuted form of ~ mpd . (Contributed by NM, 17-Mar-1996.) |
syli : |- ( ps -> ( ph -> th ) ) | theorem | set | [] | set.mm | syli | Syllogism inference with common nested antecedent. (Contributed by NM, 4-Nov-2004.) |
syl2im : |- ( ph -> ( ch -> ta ) ) | theorem | set | [] | set.mm | syl2im | Replace two antecedents. Implication-only version of ~ syl2an . (Contributed by Wolf Lammen, 14-May-2013.) |
syl2imc : |- ( ch -> ( ph -> ta ) ) | theorem | set | [] | set.mm | syl2imc | A commuted version of ~ syl2im . Implication-only version of ~ syl2anr . (Contributed by BJ, 20-Oct-2021.) |
pm2.27 : |- ( ph -> ( ( ph -> ps ) -> ps ) ) | theorem | set | [] | set.mm | pm2.27 | null |
mpdd : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | mpdd | A nested modus ponens deduction. Double deduction associated with ~ ax-mp . Deduction associated with ~ mpd . (Contributed by NM, 12-Dec-2004.) |
mpid : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | mpid | A nested modus ponens deduction. Deduction associated with ~ mpi . (Contributed by NM, 14-Dec-2004.) |
mpdi : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | mpdi | A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.) |
mpii : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | mpii | A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.) |
syld : |- ( ph -> ( ps -> th ) ) | theorem | set | [] | set.mm | syld | Syllogism deduction. Deduction associated with ~ syl . See ~ conventions for the meaning of "associated deduction" or "deduction form". (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mel L. O'Cat, 19-Feb-2008.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) |
syldc : |- ( ps -> ( ph -> th ) ) | theorem | set | [] | set.mm | syldc | Syllogism deduction. Commuted form of ~ syld . (Contributed by BJ, 25-Oct-2021.) |
mp2d : |- ( ph -> th ) | theorem | set | [] | set.mm | mp2d | null |
a1dd : |- ( ph -> ( ps -> ( th -> ch ) ) ) | theorem | set | [] | set.mm | a1dd | Double deduction introducing an antecedent. Deduction associated with ~ a1d . Double deduction associated with ~ ax-1 and ~ a1i . (Contributed by NM, 17-Dec-2004.) (Proof shortened by Mel L. O'Cat, 15-Jan-2008.) |
2a1dd : |- ( ph -> ( ps -> ( th -> ( ta -> ch ) ) ) ) | theorem | set | [] | set.mm | 2a1dd | Double deduction introducing two antecedents. Two applications of ~ 2a1dd . Deduction associated with ~ 2a1d . Double deduction associated with ~ 2a1 and ~ 2a1i . (Contributed by Jeff Hankins, 5-Aug-2009.) |
pm2.43i : |- ( ph -> ps ) | theorem | set | [] | set.mm | pm2.43i | Inference absorbing redundant antecedent. Inference associated with ~ pm2.43 . (Contributed by NM, 10-Jan-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) |
pm2.43d : |- ( ph -> ( ps -> ch ) ) | theorem | set | [] | set.mm | pm2.43d | Deduction absorbing redundant antecedent. Deduction associated with ~ pm2.43 and ~ pm2.43i . (Contributed by NM, 18-Aug-1993.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) |
pm2.43a : |- ( ps -> ( ph -> ch ) ) | theorem | set | [] | set.mm | pm2.43a | Inference absorbing redundant antecedent. (Contributed by NM, 7-Nov-1995.) (Proof shortened by Mel L. O'Cat, 28-Nov-2008.) |
pm2.43b : |- ( ph -> ( ps -> ch ) ) | theorem | set | [] | set.mm | pm2.43b | Inference absorbing redundant antecedent. (Contributed by NM, 31-Oct-1995.) |
pm2.43 : |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) | theorem | set | [] | set.mm | pm2.43 | null |
imim2d : |- ( ph -> ( ( th -> ps ) -> ( th -> ch ) ) ) | theorem | set | [] | set.mm | imim2d | Deduction adding nested antecedents. Deduction associated with ~ imim2 and ~ imim2i . (Contributed by NM, 10-Jan-1993.) |
imim2 : |- ( ( ph -> ps ) -> ( ( ch -> ph ) -> ( ch -> ps ) ) ) | theorem | set | [] | set.mm | imim2 | null |
embantd : |- ( ph -> ( ( ps -> ch ) -> th ) ) | theorem | set | [] | set.mm | embantd | Deduction embedding an antecedent. (Contributed by Wolf Lammen, 4-Oct-2013.) |
3syld : |- ( ph -> ( ps -> ta ) ) | theorem | set | [] | set.mm | 3syld | Triple syllogism deduction. Deduction associated with ~ 3syld . (Contributed by Jeff Hankins, 4-Aug-2009.) |
sylsyld : |- ( ph -> ( ch -> ta ) ) | theorem | set | [] | set.mm | sylsyld | A double syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.) |
imim12i : |- ( ( ps -> ch ) -> ( ph -> th ) ) | theorem | set | [] | set.mm | imim12i | Inference joining two implications. Inference associated with ~ imim12 . Its associated inference is ~ 3syl . (Contributed by NM, 12-Mar-1993.) (Proof shortened by Mel L. O'Cat, 29-Oct-2011.) |
imim1i : |- ( ( ps -> ch ) -> ( ph -> ch ) ) | theorem | set | [] | set.mm | imim1i | null |
imim3i : |- ( ( th -> ph ) -> ( ( th -> ps ) -> ( th -> ch ) ) ) | theorem | set | [] | set.mm | imim3i | Inference adding three nested antecedents. (Contributed by NM, 19-Dec-2006.) |
sylc : |- ( ph -> th ) | theorem | set | [] | set.mm | sylc | A syllogism inference combined with contraction. (Contributed by NM, 4-May-1994.) (Revised by NM, 13-Jul-2013.) |
syl3c : |- ( ph -> ta ) | theorem | set | [] | set.mm | syl3c | A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011.) |
syl6mpi : |- ( ph -> ( ps -> ta ) ) | theorem | set | [] | set.mm | syl6mpi | A syllogism inference. (Contributed by Alan Sare, 8-Jul-2011.) (Proof shortened by Wolf Lammen, 13-Sep-2012.) |
mpsyl : |- ( ps -> th ) | theorem | set | [] | set.mm | mpsyl | Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.) |
mpsylsyld : |- ( ps -> ( ch -> ta ) ) | theorem | set | [] | set.mm | mpsylsyld | Modus ponens combined with a double syllogism inference. (Contributed by Alan Sare, 22-Jul-2012.) |
syl6c : |- ( ph -> ( ps -> ta ) ) | theorem | set | [] | set.mm | syl6c | Inference combining ~ syl6 with contraction. (Contributed by Alan Sare, 2-May-2011.) |
syl6ci : |- ( ph -> ( ps -> ta ) ) | theorem | set | [] | set.mm | syl6ci | A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.) |
syldd : |- ( ph -> ( ps -> ( ch -> ta ) ) ) | theorem | set | [] | set.mm | syldd | Nested syllogism deduction. Deduction associated with ~ syld . Double deduction associated with ~ syl . (Contributed by NM, 12-Dec-2004.) (Proof shortened by Wolf Lammen, 11-May-2013.) |
syl5d : |- ( ph -> ( th -> ( ps -> ta ) ) ) | theorem | set | [] | set.mm | syl5d | A nested syllogism deduction. Deduction associated with ~ syl5 . (Contributed by NM, 14-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) |
syl7 : |- ( ch -> ( th -> ( ph -> ta ) ) ) | theorem | set | [] | set.mm | syl7 | A syllogism rule of inference. The first premise is used to replace the third antecedent of the second premise. (Contributed by NM, 12-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) |
syl6d : |- ( ph -> ( ps -> ( ch -> ta ) ) ) | theorem | set | [] | set.mm | syl6d | A nested syllogism deduction. Deduction associated with ~ syl6 . (Contributed by NM, 11-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by Mel L. O'Cat, 2-Feb-2006.) |
syl8 : |- ( ph -> ( ps -> ( ch -> ta ) ) ) | theorem | set | [] | set.mm | syl8 | A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.) |
syl9 : |- ( ph -> ( th -> ( ps -> ta ) ) ) | theorem | set | [] | set.mm | syl9 | A nested syllogism inference with different antecedents. (Contributed by NM, 13-May-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) |
syl9r : |- ( th -> ( ph -> ( ps -> ta ) ) ) | theorem | set | [] | set.mm | syl9r | A nested syllogism inference with different antecedents. (Contributed by NM, 14-May-1993.) |
syl10 : |- ( ph -> ( ps -> ( th -> et ) ) ) | theorem | set | [] | set.mm | syl10 | A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) |
a1ddd : |- ( ph -> ( ps -> ( ch -> ( th -> ta ) ) ) ) | theorem | set | [] | set.mm | a1ddd | Triple deduction introducing an antecedent to a wff. Deduction associated with ~ a1dd . Double deduction associated with ~ a1d . Triple deduction associated with ~ ax-1 and ~ a1i . (Contributed by Jeff Hankins, 4-Aug-2009.) |
imim12d : |- ( ph -> ( ( ch -> th ) -> ( ps -> ta ) ) ) | theorem | set | [] | set.mm | imim12d | Deduction combining antecedents and consequents. Deduction associated with ~ imim12 and ~ imim12i . (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mel L. O'Cat, 30-Oct-2011.) |
imim1d : |- ( ph -> ( ( ch -> th ) -> ( ps -> th ) ) ) | theorem | set | [] | set.mm | imim1d | Deduction adding nested consequents. Deduction associated with ~ imim1 and ~ imim1i . (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 12-Sep-2012.) |
imim1 : |- ( ( ph -> ps ) -> ( ( ps -> ch ) -> ( ph -> ch ) ) ) | theorem | set | [] | set.mm | imim1 | null |
pm2.83 : |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ( ch -> th ) ) -> ( ph -> ( ps -> th ) ) ) ) | theorem | set | [] | set.mm | pm2.83 | Theorem *2.83 of [WhiteheadRussell] p. 108. Closed form of ~ syld . (Contributed by NM, 3-Jan-2005.) |
peirceroll : |- ( ( ( ( ph -> ps ) -> ph ) -> ph ) -> ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) ) | theorem | set | [] | set.mm | peirceroll | Over minimal implicational calculus, Peirce's axiom ~ peirce implies an axiom sometimes called "Roll", ` ( ( ( ph -> ps ) -> ch ) -> ( ( ch -> ph ) -> ph ) ) ` , of which ~ looinv is a special instance. The converse also holds: substitute ` ( ph -> ps ) ` for ` ch ` in Roll and use ~ id and ~ ax-mp . (Contributed by BJ... |
com23 : |- ( ph -> ( ch -> ( ps -> th ) ) ) | theorem | set | [] | set.mm | com23 | Commutation of antecedents. Swap 2nd and 3rd. Deduction associated with ~ com12 . (Contributed by NM, 27-Dec-1992.) (Proof shortened by Wolf Lammen, 4-Aug-2012.) |
com3r : |- ( ch -> ( ph -> ( ps -> th ) ) ) | theorem | set | [] | set.mm | com3r | Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.) |
com13 : |- ( ch -> ( ps -> ( ph -> th ) ) ) | theorem | set | [] | set.mm | com13 | Commutation of antecedents. Swap 1st and 3rd. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) |
com3l : |- ( ps -> ( ch -> ( ph -> th ) ) ) | theorem | set | [] | set.mm | com3l | Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) |
pm2.04 : |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) | theorem | set | [] | set.mm | pm2.04 | null |
com34 : |- ( ph -> ( ps -> ( th -> ( ch -> ta ) ) ) ) | theorem | set | [] | set.mm | com34 | Commutation of antecedents. Swap 3rd and 4th. Deduction associated with ~ com23 . Double deduction associated with ~ com12 . (Contributed by NM, 25-Apr-1994.) |
com4l : |- ( ps -> ( ch -> ( th -> ( ph -> ta ) ) ) ) | theorem | set | [] | set.mm | com4l | Commutation of antecedents. Rotate left. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Mel L. O'Cat, 15-Aug-2004.) |
com4t : |- ( ch -> ( th -> ( ph -> ( ps -> ta ) ) ) ) | theorem | set | [] | set.mm | com4t | Commutation of antecedents. Rotate twice. (Contributed by NM, 25-Apr-1994.) |
com4r : |- ( th -> ( ph -> ( ps -> ( ch -> ta ) ) ) ) | theorem | set | [] | set.mm | com4r | Commutation of antecedents. Rotate right. (Contributed by NM, 25-Apr-1994.) |
com24 : |- ( ph -> ( th -> ( ch -> ( ps -> ta ) ) ) ) | theorem | set | [] | set.mm | com24 | Commutation of antecedents. Swap 2nd and 4th. Deduction associated with ~ com13 . (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) |
com14 : |- ( th -> ( ps -> ( ch -> ( ph -> ta ) ) ) ) | theorem | set | [] | set.mm | com14 | Commutation of antecedents. Swap 1st and 4th. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 28-Jul-2012.) |
com45 : |- ( ph -> ( ps -> ( ch -> ( ta -> ( th -> et ) ) ) ) ) | theorem | set | [] | set.mm | com45 | Commutation of antecedents. Swap 4th and 5th. Deduction associated with ~ com34 . Double deduction associated with ~ com23 . Triple deduction associated with ~ com12 . (Contributed by Jeff Hankins, 28-Jun-2009.) |
com35 : |- ( ph -> ( ps -> ( ta -> ( th -> ( ch -> et ) ) ) ) ) | theorem | set | [] | set.mm | com35 | Commutation of antecedents. Swap 3rd and 5th. Deduction associated with ~ com24 . Double deduction associated with ~ com13 . (Contributed by Jeff Hankins, 28-Jun-2009.) |
com25 : |- ( ph -> ( ta -> ( ch -> ( th -> ( ps -> et ) ) ) ) ) | theorem | set | [] | set.mm | com25 | Commutation of antecedents. Swap 2nd and 5th. Deduction associated with ~ com14 . (Contributed by Jeff Hankins, 28-Jun-2009.) |
com5l : |- ( ps -> ( ch -> ( th -> ( ta -> ( ph -> et ) ) ) ) ) | theorem | set | [] | set.mm | com5l | Commutation of antecedents. Rotate left. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.) |
com15 : |- ( ta -> ( ps -> ( ch -> ( th -> ( ph -> et ) ) ) ) ) | theorem | set | [] | set.mm | com15 | Commutation of antecedents. Swap 1st and 5th. (Contributed by Jeff Hankins, 28-Jun-2009.) (Proof shortened by Wolf Lammen, 29-Jul-2012.) |
com52l : |- ( ch -> ( th -> ( ta -> ( ph -> ( ps -> et ) ) ) ) ) | theorem | set | [] | set.mm | com52l | Commutation of antecedents. Rotate left twice. (Contributed by Jeff Hankins, 28-Jun-2009.) |
com52r : |- ( th -> ( ta -> ( ph -> ( ps -> ( ch -> et ) ) ) ) ) | theorem | set | [] | set.mm | com52r | Commutation of antecedents. Rotate right twice. (Contributed by Jeff Hankins, 28-Jun-2009.) |
com5r : |- ( ta -> ( ph -> ( ps -> ( ch -> ( th -> et ) ) ) ) ) | theorem | set | [] | set.mm | com5r | Commutation of antecedents. Rotate right. (Contributed by Wolf Lammen, 29-Jul-2012.) |
imim12 : |- ( ( ph -> ps ) -> ( ( ch -> th ) -> ( ( ps -> ch ) -> ( ph -> th ) ) ) ) | theorem | set | [] | set.mm | imim12 | Closed form of ~ imim12i and of ~ 3syl . (Contributed by BJ, 16-Jul-2019.) |
jarr : |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) | theorem | set | [] | set.mm | jarr | Elimination of a nested antecedent. Sometimes called "Syll-Simp" since it is a syllogism applied to ~ ax-1 ("Simplification"). (Contributed by Wolf Lammen, 9-May-2013.) |
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Metamath
A structured dataset of formally verified theorems and axioms from Metamath, one of the largest collections of rigorously verified mathematics in the world.
Source
- Repository: https://github.com/metamath/set.mm
- Website: https://us.metamath.org
- License: CC0 1.0 (Public Domain)
Statistics
| Property | Value |
|---|---|
| Total Entries | 72,962 |
| Theorems | 69,394 |
| Axioms | 3,568 |
| Docstring Coverage | 95.6% |
Database Distribution
| Database | Entries | Description |
|---|---|---|
| set.mm | 49,410 | Classical logic and ZFC set theory |
| iset.mm | 15,953 | Intuitionistic logic and set theory |
| nf.mm | 6,245 | New Foundations set theory |
| ql.mm | 1,176 | Quantum logic |
| hol.mm | 178 | Higher-order logic |
Schema
| Column | Type | Description |
|---|---|---|
fact |
string | Statement in form "label : assertion" |
type |
string | "theorem" or "axiom" |
library |
string | Source database (set, iset, nf, ql, hol) |
imports |
list[string] | Empty (Metamath uses single-file structure) |
filename |
string | Source .mm file |
symbolic_name |
string | Theorem/axiom label |
docstring |
string | Description with contributor attribution |
Metamath Syntax
Metamath uses a minimal syntax:
|-indicates a provable statement/\is conjunction (AND)\/is disjunction (OR)->is implication<->is biconditional-.is negationA.is universal quantifierE.is existential quantifier
Notable Theorems
The set.mm database includes proofs of major results:
- Fundamental Theorem of Calculus
- Prime Number Theorem
- Cauchy-Riemann equations
- Hahn-Banach theorem
- And thousands more...
Creator
Charles Norton (phanerozoic)
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