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Theorem cdlemedb 29390
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 28457 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 709 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
4 simpll 733 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  HL )
5 simprl 735 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  A )
6 simprr 736 . . . 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 28460 . . . 4  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
114, 5, 6, 10syl3anc 1187 . . 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 29091 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
1413ad2antlr 710 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  W  e.  B )
15 cdlemeda.m . . . 4  |-  ./\  =  ( meet `  K )
167, 15latmcl 14001 . . 3  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( R  .\/  S
)  ./\  W )  e.  B )
173, 11, 14, 16syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  B )
181, 17syl5eqel 2337 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Latclat 13995   Atomscatm 28357   HLchlt 28444   LHypclh 29077
This theorem is referenced by:  cdleme20k  29412  cdleme20l2  29414  cdleme20l  29415  cdleme20m  29416
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-lat 13996  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-lhyp 29081
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