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Theorem cdleme31sn1 29259
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme31sn1.i  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
cdleme31sn1.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
cdleme31sn1.c  |-  C  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G ) )
Assertion
Ref Expression
cdleme31sn1  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  C )
Distinct variable groups:    t, s,
y, A    B, s    .\/ , s    .<_ , s    P, s    Q, s    R, s, t, y    W, s
Allowed substitution hints:    B( y, t)    C( y, t, s)    D( y, t, s)    P( y, t)    Q( y, t)    G( y, t, s)    I( y, t, s)    .\/ ( y, t)    .<_ ( y, t)    N( y, t, s)    W( y, t)

Proof of Theorem cdleme31sn1
StepHypRef Expression
1 cdleme31sn1.n . . . 4  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
2 eqid 2253 . . . 4  |-  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )  =  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)
31, 2cdleme31sn 29258 . . 3  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  / 
s ]_ I ,  [_ R  /  s ]_ D
) )
43adantr 453 . 2  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  / 
s ]_ I ,  [_ R  /  s ]_ D
) )
5 iftrue 3476 . . . . 5  |-  ( R 
.<_  ( P  .\/  Q
)  ->  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )  =  [_ R  /  s ]_ I
)
6 cdleme31sn1.i . . . . . 6  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
76csbeq2i 3035 . . . . 5  |-  [_ R  /  s ]_ I  =  [_ R  /  s ]_ ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
85, 7syl6eq 2301 . . . 4  |-  ( R 
.<_  ( P  .\/  Q
)  ->  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )  =  [_ R  /  s ]_ ( iota_ y  e.  B A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  G ) ) )
9 nfcv 2385 . . . . . . . 8  |-  F/_ s A
10 nfv 1629 . . . . . . . . 9  |-  F/ s ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )
11 nfcsb1v 3041 . . . . . . . . . 10  |-  F/_ s [_ R  /  s ]_ G
1211nfeq2 2396 . . . . . . . . 9  |-  F/ s  y  =  [_ R  /  s ]_ G
1310, 12nfim 1735 . . . . . . . 8  |-  F/ s ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G )
149, 13nfral 2558 . . . . . . 7  |-  F/ s A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G )
15 nfcv 2385 . . . . . . 7  |-  F/_ s B
1614, 15nfriota 6200 . . . . . 6  |-  F/_ s
( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G ) )
1716a1i 12 . . . . 5  |-  ( R  e.  A  ->  F/_ s
( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G ) ) )
18 csbeq1a 3017 . . . . . . . . 9  |-  ( s  =  R  ->  G  =  [_ R  /  s ]_ G )
1918eqeq2d 2264 . . . . . . . 8  |-  ( s  =  R  ->  (
y  =  G  <->  y  =  [_ R  /  s ]_ G ) )
2019imbi2d 309 . . . . . . 7  |-  ( s  =  R  ->  (
( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G )  <->  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  [_ R  /  s ]_ G
) ) )
2120ralbidv 2527 . . . . . 6  |-  ( s  =  R  ->  ( A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G )  <->  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  [_ R  /  s ]_ G
) ) )
2221riotabidv 6192 . . . . 5  |-  ( s  =  R  ->  ( iota_ y  e.  B A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  G ) )  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G ) ) )
2317, 22csbiegf 3049 . . . 4  |-  ( R  e.  A  ->  [_ R  /  s ]_ ( iota_ y  e.  B A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  G ) )  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G ) ) )
248, 23sylan9eqr 2307 . . 3  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  [_ R  /  s ]_ G
) ) )
25 cdleme31sn1.c . . 3  |-  C  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G ) )
2624, 25syl6eqr 2303 . 2  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)  =  C )
274, 26eqtrd 2285 1  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   F/_wnfc 2372   A.wral 2509   [_csb 3009   ifcif 3470   class class class wbr 3920  (class class class)co 5710   iota_crio 6181
This theorem is referenced by:  cdleme31sn1c  29266
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-iota 6143  df-riota 6190
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