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Theorem cdleme31sn2 30883
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 2412 . . . . 5  |-  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )  =  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)
31, 2cdleme31sn 30874 . . . 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 3714 . . . . 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 3245 . . . . 5  |-  [_ R  /  s ]_ D  =  [_ R  /  s ]_ ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
85, 7syl6eq 2460 . . . 4  |-  ( -.  R  .<_  ( P  .\/  Q )  ->  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)  =  [_ R  /  s ]_ (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) ) )
9 nfcvd 2549 . . . . 5  |-  ( R  e.  A  ->  F/_ s
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
10 oveq1 6055 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  U )  =  ( R  .\/  U ) )
11 oveq2 6056 . . . . . . . 8  |-  ( s  =  R  ->  ( P  .\/  s )  =  ( P  .\/  R
) )
1211oveq1d 6063 . . . . . . 7  |-  ( s  =  R  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
1312oveq2d 6064 . . . . . 6  |-  ( s  =  R  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )
1410, 13oveq12d 6066 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
159, 14csbiegf 3259 . . . 4  |-  ( R  e.  A  ->  [_ R  /  s ]_ (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
168, 15sylan9eqr 2466 . . 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 2444 . 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 2462 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 1721   [_csb 3219   ifcif 3707   class class class wbr 4180  (class class class)co 6048
This theorem is referenced by:  cdlemefr32sn2aw  30898  cdleme43frv1snN  30902  cdlemefr31fv1  30905  cdleme35sn2aw  30952  cdleme35sn3a  30953
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051
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