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Theorem cdleme7a 30491
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30496 and cdleme7 30497. (Contributed by NM, 7-Jun-2012.)
Hypotheses
Ref Expression
cdleme4.l  |-  .<_  =  ( le `  K )
cdleme4.j  |-  .\/  =  ( join `  K )
cdleme4.m  |-  ./\  =  ( meet `  K )
cdleme4.a  |-  A  =  ( Atoms `  K )
cdleme4.h  |-  H  =  ( LHyp `  K
)
cdleme4.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme4.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme4.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
cdleme7.v  |-  V  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme7a  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  V ) )

Proof of Theorem cdleme7a
StepHypRef Expression
1 cdleme4.g . 2  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
2 cdleme7.v . . . 4  |-  V  =  ( ( R  .\/  S )  ./\  W )
32oveq2i 5992 . . 3  |-  ( F 
.\/  V )  =  ( F  .\/  (
( R  .\/  S
)  ./\  W )
)
43oveq2i 5992 . 2  |-  ( ( P  .\/  Q ) 
./\  ( F  .\/  V ) )  =  ( ( P  .\/  Q
)  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
51, 4eqtr4i 2389 1  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1647   ` cfv 5358  (class class class)co 5981   lecple 13423   joincjn 14288   meetcmee 14289   Atomscatm 29512   LHypclh 30232
This theorem is referenced by:  cdleme7d  30494  cdleme17a  30534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-iota 5322  df-fv 5366  df-ov 5984
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