Theorem ismnd 13452

Description: The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 13456), whose operation is associative (so, a semigroup, see also mndass 13457) and has a two-sided neutral element (see mndid 13458). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)

Hypotheses

Ref	Expression
ismnd.b	|- B = ( Base ` G )
ismnd.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
ismnd	|- ( G e. Mnd <-> ( A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )

Distinct variable groups:   B, a, b, c   B, e, a   G, a, b, c   .+ , a, e   .+ , b, c

Allowed substitution hint:   G( e)

Proof of Theorem ismnd

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismnd.b	. . 3 |- B = ( Base ` G )
2		ismnd.p	. . 3 |- .+ = ( +g ` G )
3	1, 2	ismnddef 13451	. 2 |- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
4		rexm 3591	. . . . 5 |- ( E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) -> E. e e e. B )
5		eleq1w 2290	. . . . . 6 |- ( e = w -> ( e e. B <-> w e. B ) )
6	5	cbvexv 1965	. . . . 5 |- ( E. e e e. B <-> E. w w e. B )
7	4, 6	sylib 122	. . . 4 |- ( E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) -> E. w w e. B )
8	1	basmex 13092	. . . . 5 |- ( w e. B -> G e. _V )
9	8	exlimiv 1644	. . . 4 |- ( E. w w e. B -> G e. _V )
10	1, 2	issgrpv 13437	. . . 4 |- ( G e. _V -> ( G e. Smgrp <-> A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) ) )
11	7, 9, 10	3syl 17	. . 3 |- ( E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) -> ( G e. Smgrp <-> A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) ) )
12	11	pm5.32ri 455	. 2 |- ( ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) <-> ( A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
13	3, 12	bitri 184	1 |- ( G e. Mnd <-> ( A. a e. B A. b e. B ( ( a .+ b ) e. B /\ A. c e. B ( ( a .+ b ) .+ c ) = ( a .+ ( b .+ c ) ) ) /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   _Vcvv 2799   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110  Smgrpcsgrp 13434   Mndcmnd 13449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6004  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-mgm 13389  df-sgrp 13435  df-mnd 13450

This theorem is referenced by:  mndid  13458  ismndd  13470  mndpropd  13473  mhmmnd  13653