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Theorem cdleme31sc 33921
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 2581 . . 3  |-  ( R  e.  A  ->  F/_ s
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
2 oveq1 6313 . . . 4  |-  ( s  =  R  ->  (
s  .\/  U )  =  ( R  .\/  U ) )
3 oveq2 6314 . . . . . 6  |-  ( s  =  R  ->  ( P  .\/  s )  =  ( P  .\/  R
) )
43oveq1d 6321 . . . . 5  |-  ( s  =  R  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
54oveq2d 6322 . . . 4  |-  ( s  =  R  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )
62, 5oveq12d 6324 . . 3  |-  ( s  =  R  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
71, 6csbiegf 3419 . 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 3812 . 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 2488 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ C  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   [_csb 3395  (class class class)co 6306
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 2057  ax-ext 2401
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 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-iota 5565  df-fv 5609  df-ov 6309
This theorem is referenced by:  cdleme31snd  33923  cdleme31sdnN  33924  cdlemefr44  33962  cdlemefr45e  33965  cdleme48fv  34036  cdleme46fvaw  34038  cdleme48bw  34039  cdleme46fsvlpq  34042  cdlemeg46fvcl  34043  cdlemeg49le  34048  cdlemeg46fjgN  34058  cdlemeg46rjgN  34059  cdlemeg46fjv  34060  cdleme48d  34072  cdlemeg49lebilem  34076  cdleme50eq  34078  cdleme50f  34079  cdlemg2jlemOLDN  34130  cdlemg2klem  34132
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