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Theorem cdleme31sn1c 30646
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 2358 . . 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 30639 . 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 30640 . . . . . . . 8  |-  ( R  e.  A  ->  [_ R  /  s ]_ G  =  Y )
87adantr 451 . . . . . . 7  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ G  =  Y )
98eqeq2d 2369 . . . . . 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 2639 . . . 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 6393 . . 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 2408 . 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 2390 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 1642    e. wcel 1710   A.wral 2619   [_csb 3157   ifcif 3641   class class class wbr 4104  (class class class)co 5945   iota_crio 6384
This theorem is referenced by:  cdlemefs32sn1aw  30672  cdleme43fsv1snlem  30678  cdleme41sn3a  30691  cdleme40m  30725  cdleme40n  30726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-ov 5948  df-riota 6391
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