Theorem ismnddef 17296

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. SGrp /\ 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		fvex 6201	. . 3 |- ( Base ` g ) e. _V
2		fvex 6201	. . 3 |- ( +g ` g ) e. _V
3		fveq2 6191	. . . . . . 7 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		ismnddef.b	. . . . . . 7 |- B = ( Base ` G )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( g = G -> ( Base ` g ) = B )
6	5	eqeq2d 2632	. . . . 5 |- ( g = G -> ( b = ( Base ` g ) <-> b = B ) )
7		fveq2 6191	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
8		ismnddef.p	. . . . . . 7 |- .+ = ( +g ` G )
9	7, 8	syl6eqr 2674	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
10	9	eqeq2d 2632	. . . . 5 |- ( g = G -> ( p = ( +g ` g ) <-> p = .+ ) )
11	6, 10	anbi12d 747	. . . 4 |- ( g = G -> ( ( b = ( Base ` g ) /\ p = ( +g ` g ) ) <-> ( b = B /\ p = .+ ) ) )
12		simpl 473	. . . . 5 |- ( ( b = B /\ p = .+ ) -> b = B )
13		oveq 6656	. . . . . . . . 9 |- ( p = .+ -> ( e p a ) = ( e .+ a ) )
14	13	eqeq1d 2624	. . . . . . . 8 |- ( p = .+ -> ( ( e p a ) = a <-> ( e .+ a ) = a ) )
15		oveq 6656	. . . . . . . . 9 |- ( p = .+ -> ( a p e ) = ( a .+ e ) )
16	15	eqeq1d 2624	. . . . . . . 8 |- ( p = .+ -> ( ( a p e ) = a <-> ( a .+ e ) = a ) )
17	14, 16	anbi12d 747	. . . . . . 7 |- ( p = .+ -> ( ( ( e p a ) = a /\ ( a p e ) = a ) <-> ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
18	17	adantl 482	. . . . . 6 |- ( ( b = B /\ p = .+ ) -> ( ( ( e p a ) = a /\ ( a p e ) = a ) <-> ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )
19	12, 18	raleqbidv 3152	. . . . 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 ) ) )
20	12, 19	rexeqbidv 3153	. . . 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 ) ) )
21	11, 20	syl6bi 243	. . 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 ) ) ) )
22	1, 2, 21	sbc2iedv 3506	. 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 ) ) )
23		df-mnd 17295	. 2 |- Mnd = { g e. SGrp | [. ( Base ` g ) / b ]. [. ( +g ` g ) / p ]. E. e e. b A. a e. b ( ( e p a ) = a /\ ( a p e ) = a ) }
24	22, 23	elrab2 3366	1 |- ( G e. Mnd <-> ( G e. SGrp /\ E. e e. B A. a e. B ( ( e .+ a ) = a /\ ( a .+ e ) = a ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   [.wsbc 3435   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  SGrpcsgrp 17283   Mndcmnd 17294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mnd 17295

This theorem is referenced by:  ismnd  17297  isnmnd  17298  mndsgrp  17299  mnd1  17331  isringrng  41881  2zrngamnd  41941