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Theorem cdleme31fv 34034
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 6056 . . . 4  |-  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) )  e.  _V
31, 2eqeltri 2513 . . 3  |-  C  e. 
_V
4 ifexg 3859 . . 3  |-  ( ( C  e.  _V  /\  X  e.  B )  ->  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )
53, 4mpan 670 . 2  |-  ( X  e.  B  ->  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )
6 breq1 4295 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
76notbid 294 . . . . 5  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
87anbi2d 703 . . . 4  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
9 oveq1 6098 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
109oveq2d 6107 . . . . . . . . . 10  |-  ( x  =  X  ->  (
s  .\/  ( x  ./\ 
W ) )  =  ( s  .\/  ( X  ./\  W ) ) )
11 id 22 . . . . . . . . . 10  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2457 . . . . . . . . 9  |-  ( x  =  X  ->  (
( s  .\/  (
x  ./\  W )
)  =  x  <->  ( s  .\/  ( X  ./\  W
) )  =  X ) )
1312anbi2d 703 . . . . . . . 8  |-  ( x  =  X  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )
149oveq2d 6107 . . . . . . . . 9  |-  ( x  =  X  ->  ( N  .\/  ( x  ./\  W ) )  =  ( N  .\/  ( X 
./\  W ) ) )
1514eqeq2d 2454 . . . . . . . 8  |-  ( x  =  X  ->  (
z  =  ( N 
.\/  ( x  ./\  W ) )  <->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
1613, 15imbi12d 320 . . . . . . 7  |-  ( x  =  X  ->  (
( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )  <->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) ) )
1716ralbidv 2735 . . . . . 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 6054 . . . . 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 2500 . . . 4  |-  ( x  =  X  ->  O  =  C )
218, 20, 11ifbieq12d 3816 . . 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 5772 . 2  |-  ( ( X  e.  B  /\  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )  ->  ( F `  X )  =  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
245, 23mpdan 668 1  |-  ( X  e.  B  ->  ( F `  X )  =  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   _Vcvv 2972   ifcif 3791   class class class wbr 4292    e. cmpt 4350   ` cfv 5418   iota_crio 6051  (class class class)co 6091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-riota 6052  df-ov 6094
This theorem is referenced by:  cdleme31fv1  34035
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