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Theorem cdleme31sc 31020
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 2572 . . 3  |-  ( R  e.  A  ->  F/_ s
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
2 oveq1 6079 . . . 4  |-  ( s  =  R  ->  (
s  .\/  U )  =  ( R  .\/  U ) )
3 oveq2 6080 . . . . . 6  |-  ( s  =  R  ->  ( P  .\/  s )  =  ( P  .\/  R
) )
43oveq1d 6087 . . . . 5  |-  ( s  =  R  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
54oveq2d 6088 . . . 4  |-  ( s  =  R  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )
62, 5oveq12d 6090 . . 3  |-  ( s  =  R  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
71, 6csbiegf 3283 . 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 3269 . 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 2492 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ C  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   [_csb 3243  (class class class)co 6072
This theorem is referenced by:  cdleme31snd  31022  cdleme31sdnN  31023  cdlemefr44  31061  cdlemefr45e  31064  cdleme48fv  31135  cdleme46fvaw  31137  cdleme48bw  31138  cdleme46fsvlpq  31141  cdlemeg46fvcl  31142  cdlemeg49le  31147  cdlemeg46fjgN  31157  cdlemeg46rjgN  31158  cdlemeg46fjv  31159  cdleme48d  31171  cdlemeg49lebilem  31175  cdleme50eq  31177  cdleme50f  31178  cdlemg2jlemOLDN  31229  cdlemg2klem  31231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5409  df-fv 5453  df-ov 6075
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