Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31sn1c Unicode version

Theorem cdleme31sn1c 30870
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 2404 . . 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 30863 . 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 30864 . . . . . . . 8  |-  ( R  e.  A  ->  [_ R  /  s ]_ G  =  Y )
87adantr 452 . . . . . . 7  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ G  =  Y )
98eqeq2d 2415 . . . . . 6  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  (
y  =  [_ R  /  s ]_ G  <->  y  =  Y ) )
109imbi2d 308 . . . . 5  |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  (
( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  [_ R  /  s ]_ G )  <->  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  Y ) ) )
1110ralbidv 2686 . . . 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 6510 . . 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 2454 . 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 2436 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 1649    e. wcel 1721   A.wral 2666   [_csb 3211   ifcif 3699   class class class wbr 4172  (class class class)co 6040   iota_crio 6501
This theorem is referenced by:  cdlemefs32sn1aw  30896  cdleme43fsv1snlem  30902  cdleme41sn3a  30915  cdleme40m  30949  cdleme40n  30950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-riota 6508
  Copyright terms: Public domain W3C validator