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Theorem grpoass 20870
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 20863 . . . 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 968 . 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 5865 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
65oveq1d 5873 . . . 4  |-  ( x  =  A  ->  (
( x G y ) G z )  =  ( ( A G y ) G z ) )
7 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x G ( y G z ) )  =  ( A G ( y G z ) ) )
86, 7eqeq12d 2297 . . 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 5866 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
109oveq1d 5873 . . . 4  |-  ( y  =  B  ->  (
( A G y ) G z )  =  ( ( A G B ) G z ) )
11 oveq1 5865 . . . . 5  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1211oveq2d 5874 . . . 4  |-  ( y  =  B  ->  ( A G ( y G z ) )  =  ( A G ( B G z ) ) )
1310, 12eqeq12d 2297 . . 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 5866 . . . 4  |-  ( z  =  C  ->  (
( A G B ) G z )  =  ( ( A G B ) G C ) )
15 oveq2 5866 . . . . 5  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
1615oveq2d 5874 . . . 4  |-  ( z  =  C  ->  ( A G ( B G z ) )  =  ( A G ( B G C ) ) )
1714, 16eqeq12d 2297 . . 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 2893 . 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 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   ran crn 4690   -->wf 5251  (class class class)co 5858   GrpOpcgr 20853
This theorem is referenced by:  grpoidinvlem1  20871  grpoidinvlem2  20872  grpoidinvlem4  20874  grporcan  20888  grpoinvid1  20897  grpoinvid2  20898  grpolcan  20900  grpo2grp  20901  grpoasscan1  20904  grpoasscan2  20905  grpoinvop  20908  grpomuldivass  20916  grponpcan  20919  grpopnpcan2  20920  gxcom  20936  gxnn0add  20941  ablo32  20953  ablo4  20954  issubgoi  20977  ghgrp  21035  rngoaass  21060  vcaass  21117  vcm  21127  nvass  21178  trran2  25393  cmprtr  25396  ltrran2  25403  ltrinvlem  25406  cmpltr2  25407  cmperltr  25409  cmprltr  25410  rltrran  25414  addvecass  25465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858
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