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Theorem cdleme31sde 29704
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 3049 . . . 4  |-  [_ S  /  t ]_ E  =  [_ S  /  t ]_ ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
3 nfcvd 2393 . . . . 5  |-  ( S  e.  A  ->  F/_ t
( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) ) )
4 oveq1 5764 . . . . . . . . 9  |-  ( t  =  S  ->  (
t  .\/  U )  =  ( S  .\/  U ) )
5 oveq2 5765 . . . . . . . . . . 11  |-  ( t  =  S  ->  ( P  .\/  t )  =  ( P  .\/  S
) )
65oveq1d 5772 . . . . . . . . . 10  |-  ( t  =  S  ->  (
( P  .\/  t
)  ./\  W )  =  ( ( P 
.\/  S )  ./\  W ) )
76oveq2d 5773 . . . . . . . . 9  |-  ( t  =  S  ->  ( Q  .\/  ( ( P 
.\/  t )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )
84, 7oveq12d 5775 . . . . . . . 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 2313 . . . . . . 7  |-  ( t  =  S  ->  D  =  Y )
12 oveq2 5765 . . . . . . . 8  |-  ( t  =  S  ->  (
s  .\/  t )  =  ( s  .\/  S ) )
1312oveq1d 5772 . . . . . . 7  |-  ( t  =  S  ->  (
( s  .\/  t
)  ./\  W )  =  ( ( s 
.\/  S )  ./\  W ) )
1411, 13oveq12d 5775 . . . . . 6  |-  ( t  =  S  ->  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) )  =  ( Y  .\/  ( ( s  .\/  S ) 
./\  W ) ) )
1514oveq2d 5773 . . . . 5  |-  ( t  =  S  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) ) )
163, 15csbiegf 3063 . . . 4  |-  ( S  e.  A  ->  [_ S  /  t ]_ (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) ) )
172, 16syl5eq 2300 . . 3  |-  ( S  e.  A  ->  [_ S  /  t ]_ E  =  ( ( P 
.\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S ) 
./\  W ) ) ) )
1817csbeq2dv 3048 . 2  |-  ( S  e.  A  ->  [_ R  /  s ]_ [_ S  /  t ]_ E  =  [_ R  /  s ]_ ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) ) )
19 eqid 2256 . . 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 29701 . 2  |-  ( R  e.  A  ->  [_ R  /  s ]_ (
( P  .\/  Q
)  ./\  ( Y  .\/  ( ( s  .\/  S )  ./\  W )
) )  =  Z )
2218, 21sylan9eqr 2310 1  |-  ( ( R  e.  A  /\  S  e.  A )  ->  [_ R  /  s ]_ [_ S  /  t ]_ E  =  Z
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   [_csb 3023  (class class class)co 5757
This theorem is referenced by:  cdlemefs44  29745  cdlemefs45ee  29749  cdleme17d2  29814
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-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fv 4654  df-ov 5760
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