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Theorem cdleme9b 36099
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 35210 . . . 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 35213 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  B )
873adant3r3 1207 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  ( P  .\/  S )  e.  B )
9 simpr3 1004 . . . 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 35844 . . . 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 15809 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  S
)  ./\  W )  e.  B )
153, 8, 12, 14syl3anc 1228 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  (
( P  .\/  S
)  ./\  W )  e.  B )
161, 15syl5eqel 2549 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 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14644   joincjn 15700   meetcmee 15701   Latclat 15802   Atomscatm 35110   HLchlt 35197   LHypclh 35830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-lat 15803  df-ats 35114  df-atl 35145  df-cvlat 35169  df-hlat 35198  df-lhyp 35834
This theorem is referenced by:  cdleme15b  36122  cdleme17b  36134
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