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Theorem cdlemk41 34399
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 2570 . 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 5825 . . . . 5  |-  ( g  =  G  ->  ( R `  g )  =  ( R `  G ) )
43oveq2d 6265 . . . 4  |-  ( g  =  G  ->  ( P  .\/  ( R `  g ) )  =  ( P  .\/  ( R `  G )
) )
5 coeq1 4954 . . . . . 6  |-  ( g  =  G  ->  (
g  o.  `' b )  =  ( G  o.  `' b ) )
65fveq2d 5829 . . . . 5  |-  ( g  =  G  ->  ( R `  ( g  o.  `' b ) )  =  ( R `  ( G  o.  `' b ) ) )
76oveq2d 6265 . . . 4  |-  ( g  =  G  ->  ( Z  .\/  ( R `  ( g  o.  `' b ) ) )  =  ( Z  .\/  ( R `  ( G  o.  `' b ) ) ) )
84, 7oveq12d 6267 . . 3  |-  ( g  =  G  ->  (
( P  .\/  ( R `  g )
)  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
92, 8syl5eq 2474 . 2  |-  ( g  =  G  ->  Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
101, 9csbiegf 3362 1  |-  ( G  e.  T  ->  [_ G  /  g ]_ Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   [_csb 3338   `'ccnv 4795    o. ccom 4800   ` cfv 5544  (class class class)co 6249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-co 4805  df-iota 5508  df-fv 5552  df-ov 6252
This theorem is referenced by:  cdlemkid2  34403  cdlemkfid3N  34404  cdlemky  34405  cdlemk42yN  34423
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