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Theorem cdleme31fv 29268
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 6194 . . . 4  |-  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) )  e.  _V
31, 2eqeltri 2323 . . 3  |-  C  e. 
_V
4 ifexg 3529 . . 3  |-  ( ( C  e.  _V  /\  X  e.  B )  ->  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )
53, 4mpan 654 . 2  |-  ( X  e.  B  ->  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )
6 breq1 3923 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
76notbid 287 . . . . 5  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
87anbi2d 687 . . . 4  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
9 oveq1 5717 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
109oveq2d 5726 . . . . . . . . . 10  |-  ( x  =  X  ->  (
s  .\/  ( x  ./\ 
W ) )  =  ( s  .\/  ( X  ./\  W ) ) )
11 id 21 . . . . . . . . . 10  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2267 . . . . . . . . 9  |-  ( x  =  X  ->  (
( s  .\/  (
x  ./\  W )
)  =  x  <->  ( s  .\/  ( X  ./\  W
) )  =  X ) )
1312anbi2d 687 . . . . . . . 8  |-  ( x  =  X  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )
149oveq2d 5726 . . . . . . . . 9  |-  ( x  =  X  ->  ( N  .\/  ( x  ./\  W ) )  =  ( N  .\/  ( X 
./\  W ) ) )
1514eqeq2d 2264 . . . . . . . 8  |-  ( x  =  X  ->  (
z  =  ( N 
.\/  ( x  ./\  W ) )  <->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
1613, 15imbi12d 313 . . . . . . 7  |-  ( x  =  X  ->  (
( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )  <->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) ) )
1716ralbidv 2527 . . . . . 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 6192 . . . . 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 2310 . . . 4  |-  ( x  =  X  ->  O  =  C )
218, 20, 11ifbieq12d 3492 . . 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 5452 . 2  |-  ( ( X  e.  B  /\  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )  ->  ( F `  X )  =  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
245, 23mpdan 652 1  |-  ( X  e.  B  ->  ( F `  X )  =  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   _Vcvv 2727   ifcif 3470   class class class wbr 3920    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   iota_crio 6181
This theorem is referenced by:  cdleme31fv1  29269
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-iota 6143  df-riota 6190
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