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Theorem cdleme43aN 30737
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 29616 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4 cdleme43.v . . . . 5  |-  V  =  ( ( Z  .\/  S )  ./\  W )
54oveq2i 5992 . . . 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 5999 . 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 2417 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 935    = wceq 1647    e. wcel 1715   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   Atomscatm 29512   HLchlt 29599   LHypclh 30232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-join 14320  df-lat 14362  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600
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