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Theorem cdleme3d 29324
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29329 and cdleme3 29330. (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 5721 . . 3  |-  ( Q 
.\/  V )  =  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
)
43oveq2i 5721 . 2  |-  ( ( R  .\/  U ) 
./\  ( Q  .\/  V ) )  =  ( ( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
51, 4eqtr4i 2276 1  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   LHypclh 29077
This theorem is referenced by:  cdleme3g  29327  cdleme3h  29328  cdleme9  29346
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 1926  ax-ext 2234
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ov 5713
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