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Theorem cdleme3d 29571
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 29576 and cdleme3 29577. (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 5789 . . 3  |-  ( Q 
.\/  V )  =  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
)
43oveq2i 5789 . 2  |-  ( ( R  .\/  U ) 
./\  ( Q  .\/  V ) )  =  ( ( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
51, 4eqtr4i 2279 1  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   ` cfv 4659  (class class class)co 5778   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   LHypclh 29324
This theorem is referenced by:  cdleme3g  29574  cdleme3h  29575  cdleme9  29593
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 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fv 4675  df-ov 5781
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