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Theorem cdlemk41 31182
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 2422 . 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 5527 . . . . 5  |-  ( g  =  G  ->  ( R `  g )  =  ( R `  G ) )
43oveq2d 5876 . . . 4  |-  ( g  =  G  ->  ( P  .\/  ( R `  g ) )  =  ( P  .\/  ( R `  G )
) )
5 coeq1 4843 . . . . . 6  |-  ( g  =  G  ->  (
g  o.  `' b )  =  ( G  o.  `' b ) )
65fveq2d 5531 . . . . 5  |-  ( g  =  G  ->  ( R `  ( g  o.  `' b ) )  =  ( R `  ( G  o.  `' b ) ) )
76oveq2d 5876 . . . 4  |-  ( g  =  G  ->  ( Z  .\/  ( R `  ( g  o.  `' b ) ) )  =  ( Z  .\/  ( R `  ( G  o.  `' b ) ) ) )
84, 7oveq12d 5878 . . 3  |-  ( g  =  G  ->  (
( P  .\/  ( R `  g )
)  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
92, 8syl5eq 2329 . 2  |-  ( g  =  G  ->  Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
101, 9csbiegf 3123 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 1625    e. wcel 1686   [_csb 3083   `'ccnv 4690    o. ccom 4695   ` cfv 5257  (class class class)co 5860
This theorem is referenced by:  cdlemkid2  31186  cdlemkfid3N  31187  cdlemky  31188  cdlemk42yN  31206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-co 4700  df-iota 5221  df-fv 5265  df-ov 5863
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