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Theorem cdleme3d 29687
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29692 and cdleme3 29693. (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 5830 . . 3  |-  ( Q 
.\/  V )  =  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
)
43oveq2i 5830 . 2  |-  ( ( R  .\/  U ) 
./\  ( Q  .\/  V ) )  =  ( ( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
51, 4eqtr4i 2307 1  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1624   ` cfv 5221  (class class class)co 5819   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   LHypclh 29440
This theorem is referenced by:  cdleme3g  29690  cdleme3h  29691  cdleme9  29709
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5822
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