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Theorem cdleme31fv 30579
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.)
Hypotheses
Ref Expression
cdleme31.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme31.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
cdleme31.c  |-  C  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
Assertion
Ref Expression
cdleme31fv  |-  ( X  e.  B  ->  ( F `  X )  =  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
Distinct variable groups:    x, B    x, C    x,  .<_    x, P    x, Q    x, W    x, s, z, X
Allowed substitution hints:    A( x, z, s)    B( z, s)    C( z, s)    P( z, s)    Q( z, s)    F( x, z, s)    .\/ ( x, z, s)    .<_ ( z, s)    ./\ ( x, z, s)    N( x, z, s)    O( x, z, s)    W( z, s)

Proof of Theorem cdleme31fv
StepHypRef Expression
1 cdleme31.c . . . 4  |-  C  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
2 riotaex 6308 . . . 4  |-  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) )  e.  _V
31, 2eqeltri 2353 . . 3  |-  C  e. 
_V
4 ifexg 3624 . . 3  |-  ( ( C  e.  _V  /\  X  e.  B )  ->  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )
53, 4mpan 651 . 2  |-  ( X  e.  B  ->  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )
6 breq1 4026 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
76notbid 285 . . . . 5  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
87anbi2d 684 . . . 4  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
9 oveq1 5865 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
109oveq2d 5874 . . . . . . . . . 10  |-  ( x  =  X  ->  (
s  .\/  ( x  ./\ 
W ) )  =  ( s  .\/  ( X  ./\  W ) ) )
11 id 19 . . . . . . . . . 10  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2297 . . . . . . . . 9  |-  ( x  =  X  ->  (
( s  .\/  (
x  ./\  W )
)  =  x  <->  ( s  .\/  ( X  ./\  W
) )  =  X ) )
1312anbi2d 684 . . . . . . . 8  |-  ( x  =  X  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )
149oveq2d 5874 . . . . . . . . 9  |-  ( x  =  X  ->  ( N  .\/  ( x  ./\  W ) )  =  ( N  .\/  ( X 
./\  W ) ) )
1514eqeq2d 2294 . . . . . . . 8  |-  ( x  =  X  ->  (
z  =  ( N 
.\/  ( x  ./\  W ) )  <->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
1613, 15imbi12d 311 . . . . . . 7  |-  ( x  =  X  ->  (
( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )  <->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) ) )
1716ralbidv 2563 . . . . . 6  |-  ( x  =  X  ->  ( A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )  <->  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
1817riotabidv 6306 . . . . 5  |-  ( x  =  X  ->  ( iota_ z  e.  B A. s  e.  A  (
( -.  s  .<_  W  /\  ( s  .\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )  =  (
iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
19 cdleme31.o . . . . 5  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
2018, 19, 13eqtr4g 2340 . . . 4  |-  ( x  =  X  ->  O  =  C )
218, 20, 11ifbieq12d 3587 . . 3  |-  ( x  =  X  ->  if ( ( P  =/= 
Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
22 cdleme31.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
2321, 22fvmptg 5600 . 2  |-  ( ( X  e.  B  /\  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )  ->  ( F `  X )  =  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
245, 23mpdan 649 1  |-  ( X  e.  B  ->  ( F `  X )  =  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297
This theorem is referenced by:  cdleme31fv1  30580
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  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-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304
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