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Theorem cdleme31sde 31119
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
Hypotheses
Ref Expression
cdleme31sde.c  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme31sde.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme31sde.x  |-  Y  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme31sde.z  |-  Z  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( R  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme31sde  |-  ( ( R  e.  A  /\  S  e.  A )  ->  [_ R  /  s ]_ [_ S  /  t ]_ E  =  Z
)
Distinct variable groups:    t, s, A    .\/ , s, t    ./\ , s,
t    P, s, t    Q, s, t    R, s    S, s, t    W, s, t    Y, s, t
Allowed substitution hints:    D( t, s)    R( t)    U( t, s)    E( t, s)    Z( t, s)

Proof of Theorem cdleme31sde
StepHypRef Expression
1 cdleme31sde.e . . . . 5  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
21csbeq2i 3269 . . . 4  |-  [_ S  /  t ]_ E  =  [_ S  /  t ]_ ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
3 nfcvd 2572 . . . . 5  |-  ( S  e.  A  ->  F/_ t
( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) ) )
4 oveq1 6080 . . . . . . . . 9  |-  ( t  =  S  ->  (
t  .\/  U )  =  ( S  .\/  U ) )
5 oveq2 6081 . . . . . . . . . . 11  |-  ( t  =  S  ->  ( P  .\/  t )  =  ( P  .\/  S
) )
65oveq1d 6088 . . . . . . . . . 10  |-  ( t  =  S  ->  (
( P  .\/  t
)  ./\  W )  =  ( ( P 
.\/  S )  ./\  W ) )
76oveq2d 6089 . . . . . . . . 9  |-  ( t  =  S  ->  ( Q  .\/  ( ( P 
.\/  t )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )
84, 7oveq12d 6091 . . . . . . . 8  |-  ( t  =  S  ->  (
( t  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
9 cdleme31sde.c . . . . . . . 8  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
10 cdleme31sde.x . . . . . . . 8  |-  Y  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
118, 9, 103eqtr4g 2492 . . . . . . 7  |-  ( t  =  S  ->  D  =  Y )
12 oveq2 6081 . . . . . . . 8  |-  ( t  =  S  ->  (
s  .\/  t )  =  ( s  .\/  S ) )
1312oveq1d 6088 . . . . . . 7  |-  ( t  =  S  ->  (
( s  .\/  t
)  ./\  W )  =  ( ( s 
.\/  S )  ./\  W ) )
1411, 13oveq12d 6091 . . . . . 6  |-  ( t  =  S  ->  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) )  =  ( Y  .\/  ( ( s  .\/  S ) 
./\  W ) ) )
1514oveq2d 6089 . . . . 5  |-  ( t  =  S  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) ) )
163, 15csbiegf 3283 . . . 4  |-  ( S  e.  A  ->  [_ S  /  t ]_ (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) ) )
172, 16syl5eq 2479 . . 3  |-  ( S  e.  A  ->  [_ S  /  t ]_ E  =  ( ( P 
.\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S ) 
./\  W ) ) ) )
1817csbeq2dv 3268 . 2  |-  ( S  e.  A  ->  [_ R  /  s ]_ [_ S  /  t ]_ E  =  [_ R  /  s ]_ ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) ) )
19 eqid 2435 . . 3  |-  ( ( P  .\/  Q ) 
./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) )  =  ( ( P  .\/  Q
)  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) )
20 cdleme31sde.z . . 3  |-  Z  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( R  .\/  S )  ./\  W )
) )
2119, 20cdleme31se 31116 . 2  |-  ( R  e.  A  ->  [_ R  /  s ]_ (
( P  .\/  Q
)  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) )  =  Z )
2218, 21sylan9eqr 2489 1  |-  ( ( R  e.  A  /\  S  e.  A )  ->  [_ R  /  s ]_ [_ S  /  t ]_ E  =  Z
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   [_csb 3243  (class class class)co 6073
This theorem is referenced by:  cdlemefs44  31160  cdlemefs45ee  31164  cdleme17d2  31229
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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