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Theorem cdleme43aN 29957
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v1) (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme43.b  |-  B  =  ( Base `  K
)
cdleme43.l  |-  .<_  =  ( le `  K )
cdleme43.j  |-  .\/  =  ( join `  K )
cdleme43.m  |-  ./\  =  ( meet `  K )
cdleme43.a  |-  A  =  ( Atoms `  K )
cdleme43.h  |-  H  =  ( LHyp `  K
)
cdleme43.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme43.x  |-  X  =  ( ( Q  .\/  P )  ./\  W )
cdleme43.c  |-  C  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme43.f  |-  Z  =  ( ( P  .\/  Q )  ./\  ( C  .\/  ( ( R  .\/  S )  ./\  W )
) )
cdleme43.d  |-  D  =  ( ( S  .\/  X )  ./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
) )
cdleme43.g  |-  G  =  ( ( Q  .\/  P )  ./\  ( D  .\/  ( ( Z  .\/  S )  ./\  W )
) )
cdleme43.e  |-  E  =  ( ( D  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  D )  ./\  W )
) )
cdleme43.v  |-  V  =  ( ( Z  .\/  S )  ./\  W )
cdleme43.y  |-  Y  =  ( ( R  .\/  D )  ./\  W )
Assertion
Ref Expression
cdleme43aN  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  G  =  ( ( P  .\/  Q ) 
./\  ( D  .\/  V ) ) )

Proof of Theorem cdleme43aN
StepHypRef Expression
1 cdleme43.j . . . 4  |-  .\/  =  ( join `  K )
2 cdleme43.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2hlatjcom 28836 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4 cdleme43.v . . . . 5  |-  V  =  ( ( Z  .\/  S )  ./\  W )
54oveq2i 5831 . . . 4  |-  ( D 
.\/  V )  =  ( D  .\/  (
( Z  .\/  S
)  ./\  W )
)
65a1i 10 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( D  .\/  V
)  =  ( D 
.\/  ( ( Z 
.\/  S )  ./\  W ) ) )
73, 6oveq12d 5838 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  .\/  Q )  ./\  ( D  .\/  V ) )  =  ( ( Q  .\/  P )  ./\  ( D  .\/  ( ( Z  .\/  S )  ./\  W )
) ) )
8 cdleme43.g . 2  |-  G  =  ( ( Q  .\/  P )  ./\  ( D  .\/  ( ( Z  .\/  S )  ./\  W )
) )
97, 8syl6reqr 2335 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  G  =  ( ( P  .\/  Q ) 
./\  ( D  .\/  V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1685   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   Atomscatm 28732   HLchlt 28819   LHypclh 29452
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-join 14106  df-lat 14148  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820
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