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Theorem cdleme31so 29847
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 2421 . . 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 5827 . . . . . . . . 9  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
32oveq2d 5836 . . . . . . . 8  |-  ( x  =  X  ->  (
s  .\/  ( x  ./\ 
W ) )  =  ( s  .\/  ( X  ./\  W ) ) )
4 id 19 . . . . . . . 8  |-  ( x  =  X  ->  x  =  X )
53, 4eqeq12d 2298 . . . . . . 7  |-  ( x  =  X  ->  (
( s  .\/  (
x  ./\  W )
)  =  x  <->  ( s  .\/  ( X  ./\  W
) )  =  X ) )
65anbi2d 684 . . . . . 6  |-  ( x  =  X  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )
72oveq2d 5836 . . . . . . 7  |-  ( x  =  X  ->  ( N  .\/  ( x  ./\  W ) )  =  ( N  .\/  ( X 
./\  W ) ) )
87eqeq2d 2295 . . . . . 6  |-  ( x  =  X  ->  (
z  =  ( N 
.\/  ( x  ./\  W ) )  <->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
96, 8imbi12d 311 . . . . 5  |-  ( x  =  X  ->  (
( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )  <->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) ) )
109ralbidv 2564 . . . 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 6302 . . 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 3122 . 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 3108 . 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 2341 1  |-  ( X  e.  B  ->  [_ X  /  x ]_ O  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   A.wral 2544   [_csb 3082   class class class wbr 4024  (class class class)co 5820   iota_crio 6291
This theorem is referenced by:  cdleme31fv1s  29860  cdlemefrs32fva  29868  cdleme32fva  29905
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 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
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 1631  df-eu 2148  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5823  df-iota 6253  df-riota 6300
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