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Theorem cmprltr2 25411
Description: Composite of two right and left translations. No restriction:  x and  z can be equal. (Contributed by FL, 2-Jul-2012.)
Hypotheses
Ref Expression
cmprltr2.1  |-  F  =  ( z  e.  X  |->  ( ( A G z ) G C ) )
cmprltr2.2  |-  E  =  ( x  e.  X  |->  ( ( B G x ) G D ) )
cmprltr2.3  |-  X  =  ran  G
Assertion
Ref Expression
cmprltr2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( F  o.  E )  =  ( x  e.  X  |->  ( ( ( A G B ) G x ) G ( D G C ) ) ) )
Distinct variable groups:    x, A    z, A    x, B    z, B    x, C    z, C    x, D    z, D    x, G    z, G    x, X    z, X
Allowed substitution hints:    E( x, z)    F( x, z)

Proof of Theorem cmprltr2
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 cmprltr2.1 . . 3  |-  F  =  ( z  e.  X  |->  ( ( A G z ) G C ) )
2 cmprltr2.2 . . . 4  |-  E  =  ( x  e.  X  |->  ( ( B G x ) G D ) )
3 oveq2 5866 . . . . . 6  |-  ( x  =  u  ->  ( B G x )  =  ( B G u ) )
43oveq1d 5873 . . . . 5  |-  ( x  =  u  ->  (
( B G x ) G D )  =  ( ( B G u ) G D ) )
54cbvmptv 4111 . . . 4  |-  ( x  e.  X  |->  ( ( B G x ) G D ) )  =  ( u  e.  X  |->  ( ( B G u ) G D ) )
62, 5eqtri 2303 . . 3  |-  E  =  ( u  e.  X  |->  ( ( B G u ) G D ) )
7 cmprltr2.3 . . 3  |-  X  =  ran  G
81, 6, 7cmprltr 25410 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( F  o.  E )  =  ( u  e.  X  |->  ( ( ( A G B ) G u ) G ( D G C ) ) ) )
9 oveq2 5866 . . . 4  |-  ( u  =  x  ->  (
( A G B ) G u )  =  ( ( A G B ) G x ) )
109oveq1d 5873 . . 3  |-  ( u  =  x  ->  (
( ( A G B ) G u ) G ( D G C ) )  =  ( ( ( A G B ) G x ) G ( D G C ) ) )
1110cbvmptv 4111 . 2  |-  ( u  e.  X  |->  ( ( ( A G B ) G u ) G ( D G C ) ) )  =  ( x  e.  X  |->  ( ( ( A G B ) G x ) G ( D G C ) ) )
128, 11syl6eq 2331 1  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X
) )  ->  ( F  o.  E )  =  ( x  e.  X  |->  ( ( ( A G B ) G x ) G ( D G C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    e. cmpt 4077   ran crn 4690    o. ccom 4693  (class class class)co 5858   GrpOpcgr 20853
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-pow 4188  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-res 4701  df-ima 4702  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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