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Theorem cdlemeulpq 29688
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-2012.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdlemeulpq  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )

Proof of Theorem cdlemeulpq
StepHypRef Expression
1 cdleme0.u . 2  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2 simpll 730 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  K  e.  HL )
3 hllat 28832 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  K  e.  Lat )
5 simprl 732 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  P  e.  A )
6 simprr 733 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  Q  e.  A )
7 eqid 2284 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme0.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdleme0.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 28835 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
12 cdleme0.h . . . . 5  |-  H  =  ( LHyp `  K
)
137, 12lhpbase 29466 . . . 4  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1413ad2antlr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  W  e.  ( Base `  K ) )
15 cdleme0.l . . . 4  |-  .<_  =  ( le `  K )
16 cdleme0.m . . . 4  |-  ./\  =  ( meet `  K )
177, 15, 16latmle1 14178 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
184, 11, 14, 17syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  -> 
( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
191, 18syl5eqbr 4057 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   Latclat 14147   Atomscatm 28732   HLchlt 28819   LHypclh 29452
This theorem is referenced by:  cdleme01N  29689  cdleme0ex1N  29691  cdleme1  29695  cdlemednuN  29768  cdleme21c  29795  cdleme22e  29812  cdleme22eALTN  29813  cdleme35fnpq  29917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-glb 14105  df-meet 14107  df-lat 14148  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-lhyp 29456
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