Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme43aN Unicode version

Theorem cdleme43aN 30751
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 29630 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
4 cdleme43.v . . . . 5  |-  V  =  ( ( Z  .\/  S )  ./\  W )
54oveq2i 5871 . . . 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 5878 . 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 2336 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 1625    e. wcel 1686   ` cfv 5257  (class class class)co 5860   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   Atomscatm 29526   HLchlt 29613   LHypclh 30246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-join 14112  df-lat 14154  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614
  Copyright terms: Public domain W3C validator