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Theorem cdleme7a 30725
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30730 and cdleme7 30731. (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 6051 . . 3  |-  ( F 
.\/  V )  =  ( F  .\/  (
( R  .\/  S
)  ./\  W )
)
43oveq2i 6051 . 2  |-  ( ( P  .\/  Q ) 
./\  ( F  .\/  V ) )  =  ( ( P  .\/  Q
)  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
51, 4eqtr4i 2427 1  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   LHypclh 30466
This theorem is referenced by:  cdleme7d  30728  cdleme17a  30768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043
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