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Theorem cdleme9b 29592
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 28704 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32adantr 453 . . 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 28707 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  B )
873adant3r3 1167 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  ( P  .\/  S )  e.  B )
9 simpr3 968 . . . 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 29338 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
129, 11syl 17 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  W  e.  B )
13 cdleme9b.m . . . 4  |-  ./\  =  ( meet `  K )
144, 13latmcl 14105 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  S
)  ./\  W )  e.  B )
153, 8, 12, 14syl3anc 1187 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  (
( P  .\/  S
)  ./\  W )  e.  B )
161, 15syl5eqel 2340 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  C  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4659  (class class class)co 5778   Basecbs 13096   joincjn 14026   meetcmee 14027   Latclat 14099   Atomscatm 28604   HLchlt 28691   LHypclh 29324
This theorem is referenced by:  cdleme15b  29615  cdleme17b  29627
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-lat 14100  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-lhyp 29328
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