Theorem ismnddef 13120

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 12761	. . . 4 |- Base Fn _V
2		vex 2766	. . . 4 |- g e. _V
3		funfvex 5578	. . . . 5 |- ( ( Fun Base /\ g e. dom Base ) -> ( Base ` g ) e. _V )
4	3	funfni 5361	. . . 4 |- ( ( Base Fn _V /\ g e. _V ) -> ( Base ` g ) e. _V )
5	1, 2, 4	mp2an 426	. . 3 |- ( Base ` g ) e. _V
6		plusgslid 12815	. . . . 5 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
7	6	slotex 12730	. . . 4 |- ( g e. _V -> ( +g ` g ) e. _V )
8	7	elv 2767	. . 3 |- ( +g ` g ) e. _V
9		fveq2 5561	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
10		ismnddef.b	. . . . . . 7 |- B = ( Base ` G )
11	9, 10	eqtr4di 2247	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
12	11	eqeq2d 2208	. . . . 5 |- ( g = G -> ( b = ( Base ` g ) <-> b = B ) )
13		fveq2 5561	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
14		ismnddef.p	. . . . . . 7 |- .+ = ( +g ` G )
15	13, 14	eqtr4di 2247	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
16	15	eqeq2d 2208	. . . . 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 5931	. . . . . . . . 9 |- ( p = .+ -> ( e p a ) = ( e .+ a ) )
20	19	eqeq1d 2205	. . . . . . . 8 |- ( p = .+ -> ( ( e p a ) = a <-> ( e .+ a ) = a ) )
21		oveq 5931	. . . . . . . . 9 |- ( p = .+ -> ( a p e ) = ( a .+ e ) )
22	21	eqeq1d 2205	. . . . . . . 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 2709	. . . . 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 2710	. . . 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 3062	. 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 13119	. 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 2923	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 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   _Vcvv 2763   [.wsbc 2989   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780  Smgrpcsgrp 13103   Mndcmnd 13118

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mnd 13119

This theorem is referenced by:  ismnd  13121  sgrpidmndm  13122  mndsgrp  13123  mnd1  13157