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Theorem cdleme31sn2 31123
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 2435 . . . . 5  |-  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )  =  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)
31, 2cdleme31sn 31114 . . . 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 3738 . . . . 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 3269 . . . . 5  |-  [_ R  /  s ]_ D  =  [_ R  /  s ]_ ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
85, 7syl6eq 2483 . . . 4  |-  ( -.  R  .<_  ( P  .\/  Q )  ->  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)  =  [_ R  /  s ]_ (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) ) )
9 nfcvd 2572 . . . . 5  |-  ( R  e.  A  ->  F/_ s
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
10 oveq1 6080 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  U )  =  ( R  .\/  U ) )
11 oveq2 6081 . . . . . . . 8  |-  ( s  =  R  ->  ( P  .\/  s )  =  ( P  .\/  R
) )
1211oveq1d 6088 . . . . . . 7  |-  ( s  =  R  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  R )  ./\  W ) )
1312oveq2d 6089 . . . . . 6  |-  ( s  =  R  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )
1410, 13oveq12d 6091 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
159, 14csbiegf 3283 . . . 4  |-  ( R  e.  A  ->  [_ R  /  s ]_ (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
168, 15sylan9eqr 2489 . . 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 2467 . 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 2485 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 1652    e. wcel 1725   [_csb 3243   ifcif 3731   class class class wbr 4204  (class class class)co 6073
This theorem is referenced by:  cdlemefr32sn2aw  31138  cdleme43frv1snN  31142  cdlemefr31fv1  31145  cdleme35sn2aw  31192  cdleme35sn3a  31193
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 5410  df-fv 5454  df-ov 6076
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