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Theorem cdlemedb 30933
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma.  D represents s2. (Contributed by NM, 20-Nov-2012.)
Hypotheses
Ref Expression
cdlemeda.l  |-  .<_  =  ( le `  K )
cdlemeda.j  |-  .\/  =  ( join `  K )
cdlemeda.m  |-  ./\  =  ( meet `  K )
cdlemeda.a  |-  A  =  ( Atoms `  K )
cdlemeda.h  |-  H  =  ( LHyp `  K
)
cdlemeda.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
cdlemedb.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdlemedb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  D  e.  B )

Proof of Theorem cdlemedb
StepHypRef Expression
1 cdlemeda.d . 2  |-  D  =  ( ( R  .\/  S )  ./\  W )
2 hllat 30000 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
4 simpll 731 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  HL )
5 simprl 733 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  A )
6 simprr 734 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  S  e.  A )
7 cdlemedb.b . . . . 5  |-  B  =  ( Base `  K
)
8 cdlemeda.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdlemeda.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 30003 . . . 4  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
114, 5, 6, 10syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( R  .\/  S
)  e.  B )
12 cdlemeda.h . . . . 5  |-  H  =  ( LHyp `  K
)
137, 12lhpbase 30634 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
1413ad2antlr 708 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  W  e.  B )
15 cdlemeda.m . . . 4  |-  ./\  =  ( meet `  K )
167, 15latmcl 14468 . . 3  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( R  .\/  S
)  ./\  W )  e.  B )
173, 11, 14, 16syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  B )
181, 17syl5eqel 2519 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  D  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Latclat 14462   Atomscatm 29900   HLchlt 29987   LHypclh 30620
This theorem is referenced by:  cdleme20k  30955  cdleme20l2  30957  cdleme20l  30958  cdleme20m  30959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-lat 14463  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-lhyp 30624
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