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Theorem cdlemk41 31109
Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 19-Jul-2013.)
Hypothesis
Ref Expression
cdlemk41.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
Assertion
Ref Expression
cdlemk41  |-  ( G  e.  T  ->  [_ G  /  g ]_ Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
Distinct variable groups:    ./\ , g    .\/ , g    g, G    P, g    R, g    T, g    g, Z   
g, b
Allowed substitution hints:    P( b)    R( b)    T( b)    G( b)    .\/ ( b)    ./\ ( b)    Y( g,
b)    Z( b)

Proof of Theorem cdlemk41
StepHypRef Expression
1 nfcvd 2420 . 2  |-  ( G  e.  T  ->  F/_ g
( ( P  .\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `  ( G  o.  `' b
) ) ) ) )
2 cdlemk41.y . . 3  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
3 fveq2 5525 . . . . 5  |-  ( g  =  G  ->  ( R `  g )  =  ( R `  G ) )
43oveq2d 5874 . . . 4  |-  ( g  =  G  ->  ( P  .\/  ( R `  g ) )  =  ( P  .\/  ( R `  G )
) )
5 coeq1 4841 . . . . . 6  |-  ( g  =  G  ->  (
g  o.  `' b )  =  ( G  o.  `' b ) )
65fveq2d 5529 . . . . 5  |-  ( g  =  G  ->  ( R `  ( g  o.  `' b ) )  =  ( R `  ( G  o.  `' b ) ) )
76oveq2d 5874 . . . 4  |-  ( g  =  G  ->  ( Z  .\/  ( R `  ( g  o.  `' b ) ) )  =  ( Z  .\/  ( R `  ( G  o.  `' b ) ) ) )
84, 7oveq12d 5876 . . 3  |-  ( g  =  G  ->  (
( P  .\/  ( R `  g )
)  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
92, 8syl5eq 2327 . 2  |-  ( g  =  G  ->  Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
101, 9csbiegf 3121 1  |-  ( G  e.  T  ->  [_ G  /  g ]_ Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   [_csb 3081   `'ccnv 4688    o. ccom 4693   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  cdlemkid2  31113  cdlemkfid3N  31114  cdlemky  31115  cdlemk42yN  31133
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  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-br 4024  df-opab 4078  df-co 4698  df-iota 5219  df-fv 5263  df-ov 5861
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