Theorem ismnddef 13631

Description: The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)

Hypotheses

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

Assertion

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

Distinct variable groups:   B, a, e   .+ , a, e

Allowed substitution hints:   G( e, a)

Proof of Theorem ismnddef

Dummy variables b g p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 13271	. . . 4 |- Base Fn _V
2		vex 2816	. . . 4 |- g e. _V
3		funfvex 5687	. . . . 5 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5458	. . . 4 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
5	1, 2, 4	mp2an 426	. . 3 |- ( Base ` g ) e. _V
6		plusgslid 13325	. . . . 5 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
7	6	slotex 13239	. . . 4 |- ( g e. _V -> ( +g ` g ) e. _V )
8	7	elv 2817	. . 3 |- ( +g ` g ) e. _V
9		fveq2 5670	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
10		ismnddef.b	. . . . . . 7 |- B = ( Base ` G )
11	9, 10	eqtr4di 2283	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
12	11	eqeq2d 2244	. . . . 5 |- ( g = G -> ( b = ( Base ` g ) <-> b = B ) )
13		fveq2 5670	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
14		ismnddef.p	. . . . . . 7 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2283	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
16	15	eqeq2d 2244	. . . . 5 |- ( g = G -> ( p = ( +g ` g ) <-> p = .+ ) )
17	12, 16	anbi12d 473	. . . 4 |- ( g = G -> ( ( b = ( Base ` g ) /\ p = ( +g ` g ) ) <-> ( b = B /\ p = .+ ) ) )
18		simpl 109	. . . . 5 |- ( ( b = B /\ p = .+ ) -> b = B )
19		oveq 6056	. . . . . . . . 9 |- ( p = .+ -> ( e p a ) = ( e .+ a ) )
20	19	eqeq1d 2241	. . . . . . . 8 |- ( p = .+ -> ( ( e p a ) = a <-> ( e .+ a ) = a ) )
21		oveq 6056	. . . . . . . . 9 |- ( p = .+ -> ( a p e ) = ( a .+ e ) )
22	21	eqeq1d 2241	. . . . . . . 8 |- ( p = .+ -> ( ( a p e ) = a <-> ( a .+ e ) = a ) )
23	20, 22	anbi12d 473	. . . . . . 7 |- ( p = .+ -> ( ( ( e p a ) = a /\ ( a p e ) = a ) <-> ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
24	23	adantl 277	. . . . . 6 |- ( ( b = B /\ p = .+ ) -> ( ( ( e p a ) = a /\ ( a p e ) = a ) <-> ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
25	18, 24	raleqbidv 2757	. . . . 5 |- ( ( b = B /\ p = .+ ) -> ( A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) <-> A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
26	18, 25	rexeqbidv 2758	. . . 4 |- ( ( b = B /\ p = .+ ) -> ( E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) <-> E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
27	17, 26	biimtrdi 163	. . 3 |- ( g = G -> ( ( b = ( Base ` g ) /\ p = ( +g ` g ) ) -> ( E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) <-> E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) ) )
28	5, 8, 27	sbc2iedv 3115	. 2 |- ( g = G -> ( [. ( Base ` g ) / b ]. [. ( +g ` g ) / p ]. E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) <-> E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
29		df-mnd 13630	. 2 |- Mnd = { g e. Smgrp | [. ( Base ` g ) / b ]. [. ( +g ` g ) / p ]. E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) }
30	28, 29	elrab2 2976	1 |- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   _Vcvv 2813   [.wsbc 3042   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290  Smgrpcsgrp 13614   Mndcmnd 13629

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-mnd 13630

This theorem is referenced by:  ismnd  13632  sgrpidmndm  13633  mndsgrp  13634  mnd1  13668