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Theorem cdleme3d 30420
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 30425 and cdleme3 30426. (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l  |-  .<_  =  ( le `  K )
cdleme1.j  |-  .\/  =  ( join `  K )
cdleme1.m  |-  ./\  =  ( meet `  K )
cdleme1.a  |-  A  =  ( Atoms `  K )
cdleme1.h  |-  H  =  ( LHyp `  K
)
cdleme1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme1.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
cdleme3.3  |-  V  =  ( ( P  .\/  R )  ./\  W )
Assertion
Ref Expression
cdleme3d  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )

Proof of Theorem cdleme3d
StepHypRef Expression
1 cdleme1.f . 2  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
2 cdleme3.3 . . . 4  |-  V  =  ( ( P  .\/  R )  ./\  W )
32oveq2i 5869 . . 3  |-  ( Q 
.\/  V )  =  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
)
43oveq2i 5869 . 2  |-  ( ( R  .\/  U ) 
./\  ( Q  .\/  V ) )  =  ( ( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
51, 4eqtr4i 2306 1  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   LHypclh 30173
This theorem is referenced by:  cdleme3g  30423  cdleme3h  30424  cdleme9  30442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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