Theorem ismnd 13680

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 13684), whose operation is associative (so, a semigroup, see also mndass 13685) and has a two-sided neutral element (see mndid 13686). (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 13679	. 2 |- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
4		rexm 3613	. . . . 5 |- ( E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) -> E. e e e. B )
5		eleq1w 2295	. . . . . 6 |- ( e = w -> ( e e. B <-> w e. B ) )
6	5	cbvexv 1970	. . . . 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 13356	. . . . 5 |- ( w e. B -> G e. _V )
9	8	exlimiv 1647	. . . 4 |- ( E. w w e. B -> G e. _V )
10	1, 2	issgrpv 13667	. . . 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 1398   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   _Vcvv 2815   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374  Smgrpcsgrp 13664   Mndcmnd 13677

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mgm 13619  df-sgrp 13665  df-mnd 13678

This theorem is referenced by:  mndid  13686  ismndd  13698  mndpropd  13701  mhmmnd  13869