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Theorem cdleme22gb 30459
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 29529 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . 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 29532 . . . 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 30391 . . . . 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 29532 . . . . . 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 30163 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2624, 25syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  W  e.  B )
277, 15latmcl 14400 . . . . 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 14399 . . . 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 14400 . . 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 2464 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 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   meetcmee 14322   Latclat 14394   Atomscatm 29429   HLchlt 29516   LHypclh 30149
This theorem is referenced by:  cdleme25a  30518  cdleme25dN  30521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-lat 14395  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-lhyp 30153
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