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Theorem cdleme31so 35468
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)
Hypotheses
Ref Expression
cdleme31so.o  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme31so.c  |-  C  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
Assertion
Ref Expression
cdleme31so  |-  ( X  e.  B  ->  [_ X  /  x ]_ O  =  C )
Distinct variable groups:    x, A    x, B    x,  .\/    x,  .<_    x,  ./\    x, N    x, s, z, X    x, W
Allowed substitution hints:    A( z, s)    B( z, s)    C( x, z, s)    .\/ ( z, s)    .<_ ( z, s)    ./\ ( z, s)    N( z, s)    O( x, z, s)    W( z, s)

Proof of Theorem cdleme31so
StepHypRef Expression
1 nfcvd 2630 . . 3  |-  ( X  e.  B  ->  F/_ x
( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
2 oveq1 6301 . . . . . . . . 9  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
32oveq2d 6310 . . . . . . . 8  |-  ( x  =  X  ->  (
s  .\/  ( x  ./\ 
W ) )  =  ( s  .\/  ( X  ./\  W ) ) )
4 id 22 . . . . . . . 8  |-  ( x  =  X  ->  x  =  X )
53, 4eqeq12d 2489 . . . . . . 7  |-  ( x  =  X  ->  (
( s  .\/  (
x  ./\  W )
)  =  x  <->  ( s  .\/  ( X  ./\  W
) )  =  X ) )
65anbi2d 703 . . . . . 6  |-  ( x  =  X  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )
72oveq2d 6310 . . . . . . 7  |-  ( x  =  X  ->  ( N  .\/  ( x  ./\  W ) )  =  ( N  .\/  ( X 
./\  W ) ) )
87eqeq2d 2481 . . . . . 6  |-  ( x  =  X  ->  (
z  =  ( N 
.\/  ( x  ./\  W ) )  <->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
96, 8imbi12d 320 . . . . 5  |-  ( x  =  X  ->  (
( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )  <->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) ) )
109ralbidv 2906 . . . 4  |-  ( 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 ) ) ) ) )
1110riotabidv 6257 . . 3  |-  ( 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 ) ) ) ) )
121, 11csbiegf 3464 . 2  |-  ( X  e.  B  ->  [_ 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 ) ) ) ) )
13 cdleme31so.o . . 3  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
1413csbeq2i 3841 . 2  |-  [_ X  /  x ]_ O  = 
[_ X  /  x ]_ ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
15 cdleme31so.c . 2  |-  C  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
1612, 14, 153eqtr4g 2533 1  |-  ( X  e.  B  ->  [_ X  /  x ]_ O  =  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   [_csb 3440   class class class wbr 4452   iota_crio 6254  (class class class)co 6294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-iota 5556  df-fv 5601  df-riota 6255  df-ov 6297
This theorem is referenced by:  cdleme31fv1s  35481  cdlemefrs32fva  35489  cdleme32fva  35526
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