Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cmprltr2 Unicode version

Theorem cmprltr2 25514
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 5882 . . . . . 6  |-  ( x  =  u  ->  ( B G x )  =  ( B G u ) )
43oveq1d 5889 . . . . 5  |-  ( x  =  u  ->  (
( B G x ) G D )  =  ( ( B G u ) G D ) )
54cbvmptv 4127 . . . 4  |-  ( x  e.  X  |->  ( ( B G x ) G D ) )  =  ( u  e.  X  |->  ( ( B G u ) G D ) )
62, 5eqtri 2316 . . 3  |-  E  =  ( u  e.  X  |->  ( ( B G u ) G D ) )
7 cmprltr2.3 . . 3  |-  X  =  ran  G
81, 6, 7cmprltr 25513 . 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 5882 . . . 4  |-  ( u  =  x  ->  (
( A G B ) G u )  =  ( ( A G B ) G x ) )
109oveq1d 5889 . . 3  |-  ( u  =  x  ->  (
( ( A G B ) G u ) G ( D G C ) )  =  ( ( ( A G B ) G x ) G ( D G C ) ) )
1110cbvmptv 4127 . 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 2344 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 1632    e. wcel 1696    e. cmpt 4093   ran crn 4706    o. ccom 4709  (class class class)co 5874   GrpOpcgr 20869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-grpo 20874
  Copyright terms: Public domain W3C validator