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Theorem grpoass 21302
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.
StepHypRef Expression
1 grpfo.1 . . . . 5  |-  X  =  ran  G
21isgrpo 21295 . . . 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 ) ) ) )
32ibi 232 . . 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 ) ) )
43simp2d 969 . 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 5988 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
65oveq1d 5996 . . . 4  |-  ( x  =  A  ->  (
( x G y ) G z )  =  ( ( A G y ) G z ) )
7 oveq1 5988 . . . 4  |-  ( x  =  A  ->  (
x G ( y G z ) )  =  ( A G ( y G z ) ) )
86, 7eqeq12d 2380 . . 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 5989 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
109oveq1d 5996 . . . 4  |-  ( y  =  B  ->  (
( A G y ) G z )  =  ( ( A G B ) G z ) )
11 oveq1 5988 . . . . 5  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1211oveq2d 5997 . . . 4  |-  ( y  =  B  ->  ( A G ( y G z ) )  =  ( A G ( B G z ) ) )
1310, 12eqeq12d 2380 . . 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 5989 . . . 4  |-  ( z  =  C  ->  (
( A G B ) G z )  =  ( ( A G B ) G C ) )
15 oveq2 5989 . . . . 5  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
1615oveq2d 5997 . . . 4  |-  ( z  =  C  ->  ( A G ( B G z ) )  =  ( A G ( B G C ) ) )
1714, 16eqeq12d 2380 . . 3  |-  ( z  =  C  ->  (
( ( A G B ) G z )  =  ( A G ( B G z ) )  <->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) ) )
188, 13, 17rspc3v 2978 . 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 ) ) ) )
194, 18mpan9 455 1  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629    X. cxp 4790   ran crn 4793   -->wf 5354  (class class class)co 5981   GrpOpcgr 21285
This theorem is referenced by:  grpoidinvlem1  21303  grpoidinvlem2  21304  grpoidinvlem4  21306  grporcan  21320  grpoinvid1  21329  grpoinvid2  21330  grpolcan  21332  grpo2grp  21333  grpoasscan1  21336  grpoasscan2  21337  grpoinvop  21340  grpomuldivass  21348  grponpcan  21351  grpopnpcan2  21352  gxcom  21368  gxnn0add  21373  ablo32  21385  ablo4  21386  issubgoi  21409  ghgrp  21467  rngoaass  21492  vcaass  21551  vcm  21561  nvass  21612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-ov 5984  df-grpo 21290
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