Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31sn1 Structured version   Unicode version

Theorem cdleme31sn1 35583
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 2467 . . . 4  |-  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )  =  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)
31, 2cdleme31sn 35582 . . 3  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  / 
s ]_ I ,  [_ R  /  s ]_ D
) )
43adantr 465 . 2  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  / 
s ]_ I ,  [_ R  /  s ]_ D
) )
5 iftrue 3951 . . . . 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 3841 . . . . 5  |-  [_ R  /  s ]_ I  =  [_ R  /  s ]_ ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
85, 7syl6eq 2524 . . . 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 2629 . . . . . . . 8  |-  F/_ s A
10 nfv 1683 . . . . . . . . 9  |-  F/ s ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )
11 nfcsb1v 3456 . . . . . . . . . 10  |-  F/_ s [_ R  /  s ]_ G
1211nfeq2 2646 . . . . . . . . 9  |-  F/ s  y  =  [_ R  /  s ]_ G
1310, 12nfim 1867 . . . . . . . 8  |-  F/ s ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G )
149, 13nfral 2853 . . . . . . 7  |-  F/ s A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G )
15 nfcv 2629 . . . . . . 7  |-  F/_ s B
1614, 15nfriota 6265 . . . . . 6  |-  F/_ s
( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G ) )
1716a1i 11 . . . . 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 3449 . . . . . . . . 9  |-  ( s  =  R  ->  G  =  [_ R  /  s ]_ G )
1918eqeq2d 2481 . . . . . . . 8  |-  ( s  =  R  ->  (
y  =  G  <->  y  =  [_ R  /  s ]_ G ) )
2019imbi2d 316 . . . . . . 7  |-  ( s  =  R  ->  (
( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G )  <->  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  [_ R  /  s ]_ G
) ) )
2120ralbidv 2906 . . . . . 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 6258 . . . . 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 3464 . . . 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 2530 . . 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 2526 . 2  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)  =  C )
274, 26eqtrd 2508 1  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   F/_wnfc 2615   A.wral 2817   [_csb 3440   ifcif 3945   class class class wbr 4453   iota_crio 6255  (class class class)co 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-riota 6256
This theorem is referenced by:  cdleme31sn1c  35590
  Copyright terms: Public domain W3C validator