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Theorem cdleme31sn1c 30577
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 1-Mar-2013.)
Hypotheses
Ref Expression
cdleme31sn1c.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme31sn1c.i  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
cdleme31sn1c.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
cdleme31sn1c.y  |-  Y  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )
cdleme31sn1c.c  |-  C  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  Y ) )
Assertion
Ref Expression
cdleme31sn1c  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  C )
Distinct variable groups:    t, s,
y, A    B, s    E, s    .\/ , s, t, y    .<_ , s, t, y    ./\ , s    P, s, t, y    Q, s, t, y    R, s, t, y    W, s
Allowed substitution hints:    B( y, t)    C( y, t, s)    D( y, t, s)    E( y, t)    G( y, t, s)    I( y, t, s)    ./\ ( y,
t)    N( y, t, s)    W( y, t)    Y( y, t, s)

Proof of Theorem cdleme31sn1c
StepHypRef Expression
1 cdleme31sn1c.i . . 3  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
2 cdleme31sn1c.n . . 3  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
3 eqid 2283 . . 3  |-  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  [_ R  /  s ]_ G
) )  =  (
iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G ) )
41, 2, 3cdleme31sn1 30570 . 2  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  [_ R  /  s ]_ G
) ) )
5 cdleme31sn1c.g . . . . . . . . 9  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
6 cdleme31sn1c.y . . . . . . . . 9  |-  Y  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t )  ./\  W
) ) )
75, 6cdleme31se 30571 . . . . . . . 8  |-  ( R  e.  A  ->  [_ R  /  s ]_ G  =  Y )
87adantr 451 . . . . . . 7  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ G  =  Y )
98eqeq2d 2294 . . . . . 6  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  (
y  =  [_ R  /  s ]_ G  <->  y  =  Y ) )
109imbi2d 307 . . . . 5  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  (
( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G )  <->  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  Y ) ) )
1110ralbidv 2563 . . . 4  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  ( A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G )  <->  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  Y ) ) )
1211riotabidv 6306 . . 3  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  ( iota_ y  e.  B A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  [_ R  /  s ]_ G ) )  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  Y ) ) )
13 cdleme31sn1c.c . . 3  |-  C  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  Y ) )
1412, 13syl6eqr 2333 . 2  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  ( iota_ y  e.  B A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  [_ R  /  s ]_ G ) )  =  C )
154, 14eqtrd 2315 1  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   [_csb 3081   ifcif 3565   class class class wbr 4023  (class class class)co 5858   iota_crio 6297
This theorem is referenced by:  cdlemefs32sn1aw  30603  cdleme43fsv1snlem  30609  cdleme41sn3a  30622  cdleme40m  30656  cdleme40n  30657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-riota 6304
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