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Theorem cdlemk41 31178
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 2495 . 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 5608 . . . . 5  |-  ( g  =  G  ->  ( R `  g )  =  ( R `  G ) )
43oveq2d 5961 . . . 4  |-  ( g  =  G  ->  ( P  .\/  ( R `  g ) )  =  ( P  .\/  ( R `  G )
) )
5 coeq1 4923 . . . . . 6  |-  ( g  =  G  ->  (
g  o.  `' b )  =  ( G  o.  `' b ) )
65fveq2d 5612 . . . . 5  |-  ( g  =  G  ->  ( R `  ( g  o.  `' b ) )  =  ( R `  ( G  o.  `' b ) ) )
76oveq2d 5961 . . . 4  |-  ( g  =  G  ->  ( Z  .\/  ( R `  ( g  o.  `' b ) ) )  =  ( Z  .\/  ( R `  ( G  o.  `' b ) ) ) )
84, 7oveq12d 5963 . . 3  |-  ( g  =  G  ->  (
( P  .\/  ( R `  g )
)  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
92, 8syl5eq 2402 . 2  |-  ( g  =  G  ->  Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
101, 9csbiegf 3197 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 1642    e. wcel 1710   [_csb 3157   `'ccnv 4770    o. ccom 4775   ` cfv 5337  (class class class)co 5945
This theorem is referenced by:  cdlemkid2  31182  cdlemkfid3N  31183  cdlemky  31184  cdlemk42yN  31202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-co 4780  df-iota 5301  df-fv 5345  df-ov 5948
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