Theorem ismnd 13709

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 13713), whose operation is associative (so, a semigroup, see also mndass 13714) and has a two-sided neutral element (see mndid 13715). (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 13708	. 2 |- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
4		rexm 3624	. . . . 5 |- ( E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) -> E. e e e. B )
5		eleq1w 2299	. . . . . 6 |- ( e = w -> ( e e. B <-> w e. B ) )
6	5	cbvexv 1974	. . . . 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 13390	. . . . 5 |- ( w e. B -> G e. _V )
9	8	exlimiv 1651	. . . 4 |- ( E. w w e. B -> G e. _V )
10	1, 2	issgrpv 13696	. . . 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 459	. 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 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   _Vcvv 2821   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408  Smgrpcsgrp 13693   Mndcmnd 13706

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-mgm 13653  df-sgrp 13694  df-mnd 13707

This theorem is referenced by:  mndid  13715  ismndd  13727  mndpropd  13730  mhmmnd  13896