Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk41 Structured version   Unicode version

Theorem cdlemk41 37043
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 2617 . 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 5848 . . . . 5  |-  ( g  =  G  ->  ( R `  g )  =  ( R `  G ) )
43oveq2d 6286 . . . 4  |-  ( g  =  G  ->  ( P  .\/  ( R `  g ) )  =  ( P  .\/  ( R `  G )
) )
5 coeq1 5149 . . . . . 6  |-  ( g  =  G  ->  (
g  o.  `' b )  =  ( G  o.  `' b ) )
65fveq2d 5852 . . . . 5  |-  ( g  =  G  ->  ( R `  ( g  o.  `' b ) )  =  ( R `  ( G  o.  `' b ) ) )
76oveq2d 6286 . . . 4  |-  ( g  =  G  ->  ( Z  .\/  ( R `  ( g  o.  `' b ) ) )  =  ( Z  .\/  ( R `  ( G  o.  `' b ) ) ) )
84, 7oveq12d 6288 . . 3  |-  ( g  =  G  ->  (
( P  .\/  ( R `  g )
)  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
92, 8syl5eq 2507 . 2  |-  ( g  =  G  ->  Y  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `
 ( G  o.  `' b ) ) ) ) )
101, 9csbiegf 3444 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 1398    e. wcel 1823   [_csb 3420   `'ccnv 4987    o. ccom 4992   ` cfv 5570  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-co 4997  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  cdlemkid2  37047  cdlemkfid3N  37048  cdlemky  37049  cdlemk42yN  37067
  Copyright terms: Public domain W3C validator