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Theorem cdleme43aN 34472
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 33351 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4 cdleme43.v . . . . 5  |-  V  =  ( ( Z  .\/  S )  ./\  W )
54oveq2i 6212 . . . 4  |-  ( D 
.\/  V )  =  ( D  .\/  (
( Z  .\/  S
)  ./\  W )
)
65a1i 11 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( D  .\/  V
)  =  ( D 
.\/  ( ( Z 
.\/  S )  ./\  W ) ) )
73, 6oveq12d 6219 . 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 2514 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  G  =  ( ( P  .\/  Q ) 
./\  ( D  .\/  V ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   Atomscatm 33247   HLchlt 33334   LHypclh 33967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-lub 15264  df-join 15266  df-lat 15336  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335
This theorem is referenced by: (None)
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