Theorem grpoass 25028

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 25021	. . . 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 1009	. 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 6302	. . . . 5 |- ( x = A -> ( x G y ) = ( A G y ) )
6	5	oveq1d 6310	. . . 4 |- ( x = A -> ( ( x G y ) G z ) = ( ( A G y ) G z ) )
7		oveq1 6302	. . . 4 |- ( x = A -> ( x G ( y G z ) ) = ( A G ( y G z ) ) )
8	6, 7	eqeq12d 2489	. . 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 6303	. . . . 5 |- ( y = B -> ( A G y ) = ( A G B ) )
10	9	oveq1d 6310	. . . 4 |- ( y = B -> ( ( A G y ) G z ) = ( ( A G B ) G z ) )
11		oveq1 6302	. . . . 5 |- ( y = B -> ( y G z ) = ( B G z ) )
12	11	oveq2d 6311	. . . 4 |- ( y = B -> ( A G ( y G z ) ) = ( A G ( B G z ) ) )
13	10, 12	eqeq12d 2489	. . 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 6303	. . . 4 |- ( z = C -> ( ( A G B ) G z ) = ( ( A G B ) G C ) )
15		oveq2 6303	. . . . 5 |- ( z = C -> ( B G z ) = ( B G C ) )
16	15	oveq2d 6311	. . . 4 |- ( z = C -> ( A G ( B G z ) ) = ( A G ( B G C ) ) )
17	14, 16	eqeq12d 2489	. . 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 3231	. 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 973   = wceq 1379   e. wcel 1767   A.wral 2817   E.wrex 2818   X. cxp 5003   ran crn 5006   -->wf 5590  (class class class)co 6295   GrpOpcgr 25011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-ov 6298  df-grpo 25016

This theorem is referenced by:  grpoidinvlem1  25029  grpoidinvlem2  25030  grpoidinvlem4  25032  grporcan  25046  grpoinvid1  25055  grpoinvid2  25056  grpolcan  25058  grpo2grp  25059  grpoasscan1  25062  grpoasscan2  25063  grpoinvop  25066  grpomuldivass  25074  grponpcan  25077  grpopnpcan2  25078  gxcom  25094  gxnn0add  25099  ablo32  25111  ablo4  25112  issubgoi  25135  ghgrp  25193  rngoaass  25218  vcaass  25277  vcm  25287  nvass  25338