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Theorem cdleme31sc 29841
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
Hypotheses
Ref Expression
cdleme31sc.c  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme31sc.x  |-  X  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
Assertion
Ref Expression
cdleme31sc  |-  ( R  e.  A  ->  [_ R  /  s ]_ C  =  X )
Distinct variable groups:    A, s    .\/ , s    ./\ , s    P, s    Q, s    R, s    U, s    W, s
Allowed substitution hints:    C( s)    X( s)

Proof of Theorem cdleme31sc
StepHypRef Expression
1 nfcvd 2422 . . 3  |-  ( R  e.  A  ->  F/_ s
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
2 oveq1 5827 . . . 4  |-  ( s  =  R  ->  (
s  .\/  U )  =  ( R  .\/  U ) )
3 oveq2 5828 . . . . . 6  |-  ( s  =  R  ->  ( P  .\/  s )  =  ( P  .\/  R
) )
43oveq1d 5835 . . . . 5  |-  ( s  =  R  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
54oveq2d 5836 . . . 4  |-  ( s  =  R  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )
62, 5oveq12d 5838 . . 3  |-  ( s  =  R  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
71, 6csbiegf 3123 . 2  |-  ( R  e.  A  ->  [_ R  /  s ]_ (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
8 cdleme31sc.c . . 3  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
98csbeq2i 3109 . 2  |-  [_ R  /  s ]_ C  =  [_ R  /  s ]_ ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
10 cdleme31sc.x . 2  |-  X  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
117, 9, 103eqtr4g 2342 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ C  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   [_csb 3083  (class class class)co 5820
This theorem is referenced by:  cdleme31snd  29843  cdleme31sdnN  29844  cdlemefr44  29882  cdlemefr45e  29885  cdleme48fv  29956  cdleme46fvaw  29958  cdleme48bw  29959  cdleme46fsvlpq  29962  cdlemeg46fvcl  29963  cdlemeg49le  29968  cdlemeg46fjgN  29978  cdlemeg46rjgN  29979  cdlemeg46fjv  29980  cdleme48d  29992  cdlemeg49lebilem  29996  cdleme50eq  29998  cdleme50f  29999  cdlemg2jlemOLDN  30050  cdlemg2klem  30052
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  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-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fv 5230  df-ov 5823
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