Theorem ismnddef 12819

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 12520	. . . 4 |- Base Fn _V
2		vex 2741	. . . 4 |- g e. _V
3		funfvex 5533	. . . . 5 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5317	. . . 4 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
5	1, 2, 4	mp2an 426	. . 3 |- ( Base ` g ) e. _V
6		plusgslid 12571	. . . . 5 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
7	6	slotex 12489	. . . 4 |- ( g e. _V -> ( +g ` g ) e. _V )
8	7	elv 2742	. . 3 |- ( +g ` g ) e. _V
9		fveq2 5516	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
10		ismnddef.b	. . . . . . 7 |- B = ( Base ` G )
11	9, 10	eqtr4di 2228	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
12	11	eqeq2d 2189	. . . . 5 |- ( g = G -> ( b = ( Base ` g ) <-> b = B ) )
13		fveq2 5516	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
14		ismnddef.p	. . . . . . 7 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2228	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
16	15	eqeq2d 2189	. . . . 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 5881	. . . . . . . . 9 |- ( p = .+ -> ( e p a ) = ( e .+ a ) )
20	19	eqeq1d 2186	. . . . . . . 8 |- ( p = .+ -> ( ( e p a ) = a <-> ( e .+ a ) = a ) )
21		oveq 5881	. . . . . . . . 9 |- ( p = .+ -> ( a p e ) = ( a .+ e ) )
22	21	eqeq1d 2186	. . . . . . . 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 2685	. . . . 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 2686	. . . 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	syl6bi 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 3036	. 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 12818	. 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 2897	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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2738   [.wsbc 2963   Fn wfn 5212   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536  Smgrpcsgrp 12807   Mndcmnd 12817

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-ov 5878  df-inn 8920  df-2 8978  df-ndx 12465  df-slot 12466  df-base 12468  df-plusg 12549  df-mnd 12818

This theorem is referenced by:  ismnd  12820  sgrpidmndm  12821  mndsgrp  12822  mnd1  12847