Theorem issgrp 17285

Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by by AV, 6-Jan-2020.)

Hypotheses

Ref	Expression
issgrp.b	|- B = ( Base ` M )
issgrp.o	|- .o. = ( +g ` M )

Assertion

Ref	Expression
issgrp	|- ( M e. SGrp <-> ( M e. Mgm /\ A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )

Distinct variable groups:   x, B, y, z   x, M, y, z   x, .o. , y, z

Proof of Theorem issgrp

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

Step	Hyp	Ref	Expression
1		fvexd 6203	. . 3 |- ( g = M -> ( Base ` g ) e. _V )
2		fveq2 6191	. . . 4 |- ( g = M -> ( Base ` g ) = ( Base ` M ) )
3		issgrp.b	. . . 4 |- B = ( Base ` M )
4	2, 3	syl6eqr 2674	. . 3 |- ( g = M -> ( Base ` g ) = B )
5		fvexd 6203	. . . 4 |- ( ( g = M /\ b = B ) -> ( +g ` g ) e. _V )
6		fveq2 6191	. . . . . 6 |- ( g = M -> ( +g ` g ) = ( +g ` M ) )
7	6	adantr 481	. . . . 5 |- ( ( g = M /\ b = B ) -> ( +g ` g ) = ( +g ` M ) )
8		issgrp.o	. . . . 5 |- .o. = ( +g ` M )
9	7, 8	syl6eqr 2674	. . . 4 |- ( ( g = M /\ b = B ) -> ( +g ` g ) = .o. )
10		simplr 792	. . . . 5 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> b = B )
11		id 22	. . . . . . . . . 10 |- ( o = .o. -> o = .o. )
12		oveq 6656	. . . . . . . . . 10 |- ( o = .o. -> ( x o y ) = ( x .o. y ) )
13		eqidd 2623	. . . . . . . . . 10 |- ( o = .o. -> z = z )
14	11, 12, 13	oveq123d 6671	. . . . . . . . 9 |- ( o = .o. -> ( ( x o y ) o z ) = ( ( x .o. y ) .o. z ) )
15		eqidd 2623	. . . . . . . . . 10 |- ( o = .o. -> x = x )
16		oveq 6656	. . . . . . . . . 10 |- ( o = .o. -> ( y o z ) = ( y .o. z ) )
17	11, 15, 16	oveq123d 6671	. . . . . . . . 9 |- ( o = .o. -> ( x o ( y o z ) ) = ( x .o. ( y .o. z ) ) )
18	14, 17	eqeq12d 2637	. . . . . . . 8 |- ( o = .o. -> ( ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
19	18	adantl 482	. . . . . . 7 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> ( ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
20	10, 19	raleqbidv 3152	. . . . . 6 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> ( A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
21	10, 20	raleqbidv 3152	. . . . 5 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> ( A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
22	10, 21	raleqbidv 3152	. . . 4 |- ( ( ( g = M /\ b = B ) /\ o = .o. ) -> ( A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
23	5, 9, 22	sbcied2 3473	. . 3 |- ( ( g = M /\ b = B ) -> ( [. ( +g ` g ) / o ]. A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
24	1, 4, 23	sbcied2 3473	. 2 |- ( g = M -> ( [. ( Base ` g ) / b ]. [. ( +g ` g ) / o ]. A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) <-> A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )
25		df-sgrp 17284	. 2 |- SGrp = { g e. Mgm | [. ( Base ` g ) / b ]. [. ( +g ` g ) / o ]. A. x e. b A. y e. b A. z e. b ( ( x o y ) o z ) = ( x o ( y o z ) ) }
26	24, 25	elrab2 3366	1 |- ( M e. SGrp <-> ( M e. Mgm /\ A. x e. B A. y e. B A. z e. B ( ( x .o. y ) .o. z ) = ( x .o. ( y .o. z ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Mgmcmgm 17240  SGrpcsgrp 17283

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-sgrp 17284

This theorem is referenced by:  issgrpv  17286  issgrpn0  17287  isnsgrp  17288  sgrpmgm  17289  sgrpass  17290  sgrp0  17291  sgrp0b  17292  sgrp1  17293  sgrp2nmndlem4  17415  copissgrp  41808  nnsgrp  41817  sgrpplusgaopALT  41831  sgrp2sgrp  41864  lidlmsgrp  41926  2zrngasgrp  41940  2zrngmsgrp  41947