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Theorem cdlemedb 30486
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 29553 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 706 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
4 simpll 730 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  HL )
5 simprl 732 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  A )
6 simprr 733 . . . 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 29556 . . . 4  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
114, 5, 6, 10syl3anc 1182 . . 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 30187 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
1413ad2antlr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  W  e.  B )
15 cdlemeda.m . . . 4  |-  ./\  =  ( meet `  K )
167, 15latmcl 14157 . . 3  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( R  .\/  S
)  ./\  W )  e.  B )
173, 11, 14, 16syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  B )
181, 17syl5eqel 2367 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 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme20k  30508  cdleme20l2  30510  cdleme20l  30511  cdleme20m  30512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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