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Theorem grpoass 21796
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 21789 . . . 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 234 . . 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 971 . 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 6091 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
65oveq1d 6099 . . . 4  |-  ( x  =  A  ->  (
( x G y ) G z )  =  ( ( A G y ) G z ) )
7 oveq1 6091 . . . 4  |-  ( x  =  A  ->  (
x G ( y G z ) )  =  ( A G ( y G z ) ) )
86, 7eqeq12d 2452 . . 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 6092 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
109oveq1d 6099 . . . 4  |-  ( y  =  B  ->  (
( A G y ) G z )  =  ( ( A G B ) G z ) )
11 oveq1 6091 . . . . 5  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
1211oveq2d 6100 . . . 4  |-  ( y  =  B  ->  ( A G ( y G z ) )  =  ( A G ( B G z ) ) )
1310, 12eqeq12d 2452 . . 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 6092 . . . 4  |-  ( z  =  C  ->  (
( A G B ) G z )  =  ( ( A G B ) G C ) )
15 oveq2 6092 . . . . 5  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
1615oveq2d 6100 . . . 4  |-  ( z  =  C  ->  ( A G ( B G z ) )  =  ( A G ( B G C ) ) )
1714, 16eqeq12d 2452 . . 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 3063 . 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 457 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    X. cxp 4879   ran crn 4882   -->wf 5453  (class class class)co 6084   GrpOpcgr 21779
This theorem is referenced by:  grpoidinvlem1  21797  grpoidinvlem2  21798  grpoidinvlem4  21800  grporcan  21814  grpoinvid1  21823  grpoinvid2  21824  grpolcan  21826  grpo2grp  21827  grpoasscan1  21830  grpoasscan2  21831  grpoinvop  21834  grpomuldivass  21842  grponpcan  21845  grpopnpcan2  21846  gxcom  21862  gxnn0add  21867  ablo32  21879  ablo4  21880  issubgoi  21903  ghgrp  21961  rngoaass  21986  vcaass  22045  vcm  22055  nvass  22106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-ov 6087  df-grpo 21784
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