Theorem grpoass 23827

Description: A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpoass	|- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( A G ( B G C ) ) )

Proof of Theorem grpoass

Dummy variables x y z u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpfo.1	. . . . 5 |- X = ran G
2	1	isgrpo 23820	. . . 4 |- ( G e. GrpOp -> ( G e. GrpOp <-> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) /\ E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) ) ) )
3	2	ibi 241	. . 3 |- ( G e. GrpOp -> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) /\ E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) ) )
4	3	simp2d 1001	. 2 |- ( G e. GrpOp -> A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) )
5		oveq1 6199	. . . . 5 |- ( x = A -> ( x G y ) = ( A G y ) )
6	5	oveq1d 6207	. . . 4 |- ( x = A -> ( ( x G y ) G z ) = ( ( A G y ) G z ) )
7		oveq1 6199	. . . 4 |- ( x = A -> ( x G ( y G z ) ) = ( A G ( y G z ) ) )
8	6, 7	eqeq12d 2473	. . 3 |- ( x = A -> ( ( ( x G y ) G z ) = ( x G ( y G z ) ) <-> ( ( A G y ) G z ) = ( A G ( y G z ) ) ) )
9		oveq2 6200	. . . . 5 |- ( y = B -> ( A G y ) = ( A G B ) )
10	9	oveq1d 6207	. . . 4 |- ( y = B -> ( ( A G y ) G z ) = ( ( A G B ) G z ) )
11		oveq1 6199	. . . . 5 |- ( y = B -> ( y G z ) = ( B G z ) )
12	11	oveq2d 6208	. . . 4 |- ( y = B -> ( A G ( y G z ) ) = ( A G ( B G z ) ) )
13	10, 12	eqeq12d 2473	. . 3 |- ( y = B -> ( ( ( A G y ) G z ) = ( A G ( y G z ) ) <-> ( ( A G B ) G z ) = ( A G ( B G z ) ) ) )
14		oveq2 6200	. . . 4 |- ( z = C -> ( ( A G B ) G z ) = ( ( A G B ) G C ) )
15		oveq2 6200	. . . . 5 |- ( z = C -> ( B G z ) = ( B G C ) )
16	15	oveq2d 6208	. . . 4 |- ( z = C -> ( A G ( B G z ) ) = ( A G ( B G C ) ) )
17	14, 16	eqeq12d 2473	. . 3 |- ( z = C -> ( ( ( A G B ) G z ) = ( A G ( B G z ) ) <-> ( ( A G B ) G C ) = ( A G ( B G C ) ) ) )
18	8, 13, 17	rspc3v 3181	. 2 |- ( ( A e. X /\ B e. X /\ C e. X ) -> ( A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) -> ( ( A G B ) G C ) = ( A G ( B G C ) ) ) )
19	4, 18	mpan9 469	1 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( A G ( B G C ) ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2795   E.wrex 2796   X. cxp 4938   ran crn 4941   -->wf 5514  (class class class)co 6192   GrpOpcgr 23810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524  df-fv 5526  df-ov 6195  df-grpo 23815

This theorem is referenced by:  grpoidinvlem1  23828  grpoidinvlem2  23829  grpoidinvlem4  23831  grporcan  23845  grpoinvid1  23854  grpoinvid2  23855  grpolcan  23857  grpo2grp  23858  grpoasscan1  23861  grpoasscan2  23862  grpoinvop  23865  grpomuldivass  23873  grponpcan  23876  grpopnpcan2  23877  gxcom  23893  gxnn0add  23898  ablo32  23910  ablo4  23911  issubgoi  23934  ghgrp  23992  rngoaass  24017  vcaass  24076  vcm  24086  nvass  24137