Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme22gb Unicode version

Theorem cdleme22gb 29750
Description: Utility lemma for Lemma E in [Crawley] p. 115. (Contributed by NM, 5-Dec-2012.)
Hypotheses
Ref Expression
cdleme18d.l  |-  .<_  =  ( le `  K )
cdleme18d.j  |-  .\/  =  ( join `  K )
cdleme18d.m  |-  ./\  =  ( meet `  K )
cdleme18d.a  |-  A  =  ( Atoms `  K )
cdleme18d.h  |-  H  =  ( LHyp `  K
)
cdleme18d.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme18d.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme18d.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
cdleme22.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdleme22gb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  G  e.  B )

Proof of Theorem cdleme22gb
StepHypRef Expression
1 cdleme18d.g . 2  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
2 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  K  e.  HL )
3 hllat 28820 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  K  e.  Lat )
5 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  P  e.  A )
6 simp2r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  Q  e.  A )
7 cdleme22.b . . . . 5  |-  B  =  ( Base `  K
)
8 cdleme18d.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdleme18d.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 28823 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
112, 5, 6, 10syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( P  .\/  Q )  e.  B
)
12 simp1 957 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simp3r 986 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  S  e.  A )
14 cdleme18d.l . . . . . 6  |-  .<_  =  ( le `  K )
15 cdleme18d.m . . . . . 6  |-  ./\  =  ( meet `  K )
16 cdleme18d.h . . . . . 6  |-  H  =  ( LHyp `  K
)
17 cdleme18d.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme18d.f . . . . . 6  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
1914, 8, 15, 9, 16, 17, 18, 7cdleme1b 29682 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A ) )  ->  F  e.  B )
2012, 5, 6, 13, 19syl13anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  F  e.  B )
21 simp3l 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  R  e.  A )
227, 8, 9hlatjcl 28823 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
232, 21, 13, 22syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( R  .\/  S )  e.  B
)
24 simp1r 982 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  W  e.  H )
257, 16lhpbase 29454 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2624, 25syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  W  e.  B )
277, 15latmcl 14151 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( R  .\/  S
)  ./\  W )  e.  B )
284, 23, 26, 27syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( ( R  .\/  S )  ./\  W )  e.  B )
297, 8latjcl 14150 . . . 4  |-  ( ( K  e.  Lat  /\  F  e.  B  /\  ( ( R  .\/  S )  ./\  W )  e.  B )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  B
)
304, 20, 28, 29syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( F  .\/  ( ( R  .\/  S )  ./\  W )
)  e.  B )
317, 15latmcl 14151 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  B
)  ->  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S ) 
./\  W ) ) )  e.  B )
324, 11, 30, 31syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S ) 
./\  W ) ) )  e.  B )
331, 32syl5eqel 2368 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  G  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Latclat 14145   Atomscatm 28720   HLchlt 28807   LHypclh 29440
This theorem is referenced by:  cdleme25a  29809  cdleme25dN  29812
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  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-fv 5229  df-ov 5822  df-lat 14146  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-lhyp 29444
  Copyright terms: Public domain W3C validator