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Theorem cdleme31sn2 30504
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme32sn2.d  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme31sn2.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
cdleme31sn2.c  |-  C  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
Assertion
Ref Expression
cdleme31sn2  |-  ( ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  [_ R  /  s ]_ N  =  C
)
Distinct variable groups:    A, s    .\/ , s    .<_ , s    ./\ , s    P, s    Q, s    R, s    U, s    W, s
Allowed substitution hints:    C( s)    D( s)    I( s)    N( s)

Proof of Theorem cdleme31sn2
StepHypRef Expression
1 cdleme31sn2.n . . . . 5  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
2 eqid 2388 . . . . 5  |-  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )  =  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)
31, 2cdleme31sn 30495 . . . 4  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  / 
s ]_ I ,  [_ R  /  s ]_ D
) )
43adantr 452 . . 3  |-  ( ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  [_ R  /  s ]_ N  =  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
) )
5 iffalse 3690 . . . . 5  |-  ( -.  R  .<_  ( P  .\/  Q )  ->  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)  =  [_ R  /  s ]_ D
)
6 cdleme32sn2.d . . . . . 6  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
76csbeq2i 3221 . . . . 5  |-  [_ R  /  s ]_ D  =  [_ R  /  s ]_ ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
85, 7syl6eq 2436 . . . 4  |-  ( -.  R  .<_  ( P  .\/  Q )  ->  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)  =  [_ R  /  s ]_ (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) ) )
9 nfcvd 2525 . . . . 5  |-  ( R  e.  A  ->  F/_ s
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
10 oveq1 6028 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  U )  =  ( R  .\/  U ) )
11 oveq2 6029 . . . . . . . 8  |-  ( s  =  R  ->  ( P  .\/  s )  =  ( P  .\/  R
) )
1211oveq1d 6036 . . . . . . 7  |-  ( s  =  R  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
1312oveq2d 6037 . . . . . 6  |-  ( s  =  R  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )
1410, 13oveq12d 6039 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
159, 14csbiegf 3235 . . . 4  |-  ( R  e.  A  ->  [_ R  /  s ]_ (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
168, 15sylan9eqr 2442 . . 3  |-  ( ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  if ( R 
.<_  ( P  .\/  Q
) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )  =  ( ( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
174, 16eqtrd 2420 . 2  |-  ( ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  [_ R  /  s ]_ N  =  (
( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
18 cdleme31sn2.c . 2  |-  C  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
1917, 18syl6eqr 2438 1  |-  ( ( R  e.  A  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  [_ R  /  s ]_ N  =  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   [_csb 3195   ifcif 3683   class class class wbr 4154  (class class class)co 6021
This theorem is referenced by:  cdlemefr32sn2aw  30519  cdleme43frv1snN  30523  cdlemefr31fv1  30526  cdleme35sn2aw  30573  cdleme35sn3a  30574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024
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