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Theorem cdleme9b 33736
Description: Utility lemma for Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Oct-2012.)
Hypotheses
Ref Expression
cdleme9b.b  |-  B  =  ( Base `  K
)
cdleme9b.j  |-  .\/  =  ( join `  K )
cdleme9b.m  |-  ./\  =  ( meet `  K )
cdleme9b.a  |-  A  =  ( Atoms `  K )
cdleme9b.h  |-  H  =  ( LHyp `  K
)
cdleme9b.c  |-  C  =  ( ( P  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme9b  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  C  e.  B )

Proof of Theorem cdleme9b
StepHypRef Expression
1 cdleme9b.c . 2  |-  C  =  ( ( P  .\/  S )  ./\  W )
2 hllat 32848 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32adantr 465 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  K  e.  Lat )
4 cdleme9b.b . . . . 5  |-  B  =  ( Base `  K
)
5 cdleme9b.j . . . . 5  |-  .\/  =  ( join `  K )
6 cdleme9b.a . . . . 5  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 32851 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  B )
873adant3r3 1198 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  ( P  .\/  S )  e.  B )
9 simpr3 996 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  W  e.  H )
10 cdleme9b.h . . . . 5  |-  H  =  ( LHyp `  K
)
114, 10lhpbase 33482 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
129, 11syl 16 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  W  e.  B )
13 cdleme9b.m . . . 4  |-  ./\  =  ( meet `  K )
144, 13latmcl 15214 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  S
)  ./\  W )  e.  B )
153, 8, 12, 14syl3anc 1218 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  (
( P  .\/  S
)  ./\  W )  e.  B )
161, 15syl5eqel 2522 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  C  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   Basecbs 14166   joincjn 15106   meetcmee 15107   Latclat 15207   Atomscatm 32748   HLchlt 32835   LHypclh 33468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-lat 15208  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-lhyp 33472
This theorem is referenced by:  cdleme15b  33759  cdleme17b  33771
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