Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cmpltr2 Unicode version

Theorem cmpltr2 25407
Description: Composite of two left translations. The terms  A and  B are constant. (Contributed by FL, 2-Jul-2012.)
Hypotheses
Ref Expression
ltrdom.1  |-  F  =  ( x  e.  X  |->  ( A G x ) )
ltrdom.2  |-  X  =  ran  G
cmpltr2.2  |-  H  =  ( x  e.  X  |->  ( B G x ) )
Assertion
Ref Expression
cmpltr2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( ( A G B ) G x ) ) )
Distinct variable groups:    x, A    x, B    x, G    x, X
Allowed substitution hints:    F( x)    H( x)

Proof of Theorem cmpltr2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ltrdom.2 . . . . . 6  |-  X  =  ran  G
21grpocl 20867 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  z  e.  X )  ->  ( B G z )  e.  X )
323expa 1151 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  B  e.  X )  /\  z  e.  X
)  ->  ( B G z )  e.  X )
433adantl2 1112 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  z  e.  X
)  ->  ( B G z )  e.  X )
5 cmpltr2.2 . . . . 5  |-  H  =  ( x  e.  X  |->  ( B G x ) )
6 oveq2 5866 . . . . . 6  |-  ( x  =  z  ->  ( B G x )  =  ( B G z ) )
76cbvmptv 4111 . . . . 5  |-  ( x  e.  X  |->  ( B G x ) )  =  ( z  e.  X  |->  ( B G z ) )
85, 7eqtri 2303 . . . 4  |-  H  =  ( z  e.  X  |->  ( B G z ) )
98a1i 10 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  H  =  ( z  e.  X  |->  ( B G z ) ) )
10 ltrdom.1 . . . 4  |-  F  =  ( x  e.  X  |->  ( A G x ) )
1110a1i 10 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F  =  ( x  e.  X  |->  ( A G x ) ) )
12 oveq2 5866 . . 3  |-  ( x  =  ( B G z )  ->  ( A G x )  =  ( A G ( B G z ) ) )
134, 9, 11, 12fmptco 5691 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( z  e.  X  |->  ( A G ( B G z ) ) ) )
141grpoass 20870 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  x  e.  X )
)  ->  ( ( A G B ) G x )  =  ( A G ( B G x ) ) )
15143exp2 1169 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  (
x  e.  X  -> 
( ( A G B ) G x )  =  ( A G ( B G x ) ) ) ) ) )
16153imp1 1164 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  x  e.  X
)  ->  ( ( A G B ) G x )  =  ( A G ( B G x ) ) )
1716mpteq2dva 4106 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
x  e.  X  |->  ( ( A G B ) G x ) )  =  ( x  e.  X  |->  ( A G ( B G x ) ) ) )
186oveq2d 5874 . . . 4  |-  ( x  =  z  ->  ( A G ( B G x ) )  =  ( A G ( B G z ) ) )
1918cbvmptv 4111 . . 3  |-  ( x  e.  X  |->  ( A G ( B G x ) ) )  =  ( z  e.  X  |->  ( A G ( B G z ) ) )
2017, 19syl6req 2332 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
z  e.  X  |->  ( A G ( B G z ) ) )  =  ( x  e.  X  |->  ( ( A G B ) G x ) ) )
2113, 20eqtrd 2315 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( ( A G B ) G x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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 is referenced by:  cmpltr  25408
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
  Copyright terms: Public domain W3C validator