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Theorem cdlemedb 34260
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 33327 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 725 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
4 simpll 753 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  HL )
5 simprl 755 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  A )
6 simprr 756 . . . 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 33330 . . . 4  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
114, 5, 6, 10syl3anc 1219 . . 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 33961 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
1413ad2antlr 726 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  W  e.  B )
15 cdlemeda.m . . . 4  |-  ./\  =  ( meet `  K )
167, 15latmcl 15336 . . 3  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( R  .\/  S
)  ./\  W )  e.  B )
173, 11, 14, 16syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  B )
181, 17syl5eqel 2544 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  D  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   Latclat 15329   Atomscatm 33227   HLchlt 33314   LHypclh 33947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-lat 15330  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-lhyp 33951
This theorem is referenced by:  cdleme20k  34282  cdleme20l2  34284  cdleme20l  34285  cdleme20m  34286
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