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Theorem cdleme43aN 36631
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 35508 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4 cdleme43.v . . . . 5  |-  V  =  ( ( Z  .\/  S )  ./\  W )
54oveq2i 6281 . . . 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 6288 . 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 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491   LHypclh 36124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-lub 15806  df-join 15808  df-lat 15878  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492
This theorem is referenced by: (None)
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