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Theorem cdleme7a 30729
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 30734 and cdleme7 30735. (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 6055 . . 3  |-  ( F 
.\/  V )  =  ( F  .\/  (
( R  .\/  S
)  ./\  W )
)
43oveq2i 6055 . 2  |-  ( ( P  .\/  Q ) 
./\  ( F  .\/  V ) )  =  ( ( P  .\/  Q
)  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
51, 4eqtr4i 2431 1  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   LHypclh 30470
This theorem is referenced by:  cdleme7d  30732  cdleme17a  30772
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 2389
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-iota 5381  df-fv 5425  df-ov 6047
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