Theorem ismnd 13003

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 13007), whose operation is associative (so, a semigroup, see also mndass 13008) and has a two-sided neutral element (see mndid 13009). (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 13002	. 2 |- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
4		rexm 3547	. . . . 5 |- ( E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) -> E. e e e. B )
5		eleq1w 2254	. . . . . 6 |- ( e = w -> ( e e. B <-> w e. B ) )
6	5	cbvexv 1930	. . . . 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 12680	. . . . 5 |- ( w e. B -> G e. _V )
9	8	exlimiv 1609	. . . 4 |- ( E. w w e. B -> G e. _V )
10	1, 2	issgrpv 12990	. . . 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 1364   E.wex 1503   e. wcel 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698  Smgrpcsgrp 12987   Mndcmnd 13000

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-mgm 12942  df-sgrp 12988  df-mnd 13001

This theorem is referenced by:  mndid  13009  ismndd  13021  mndpropd  13024  mhmmnd  13189