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Theorem cdleme0aa 29678
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-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 )
cdleme0.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdleme0aa  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  B )

Proof of Theorem cdleme0aa
StepHypRef Expression
1 cdleme0.u . 2  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2 simp1l 979 . . . 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 cdleme0.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cdleme0.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 28758 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
873ad2ant2 977 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  P  e.  B )
95, 6atbase 28758 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
1093ad2ant3 978 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  Q  e.  B )
11 cdleme0.j . . . . 5  |-  .\/  =  ( join `  K )
125, 11latjcl 14152 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
134, 8, 10, 12syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  ( P  .\/  Q )  e.  B
)
14 simp1r 980 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  W  e.  H )
15 cdleme0.h . . . . 5  |-  H  =  ( LHyp `  K
)
165, 15lhpbase 29466 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
1714, 16syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  W  e.  B )
18 cdleme0.m . . . 4  |-  ./\  =  ( meet `  K )
195, 18latmcl 14153 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  Q
)  ./\  W )  e.  B )
204, 13, 17, 19syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  B )
211, 20syl5eqel 2368 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   ` 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:  cdleme1b  29694  cdleme5  29708  cdleme9  29721  cdleme11g  29733  cdleme11  29738  cdleme35fnpq  29917  cdleme42e  29947  cdlemeg46frv  29993  cdlemg2fv2  30068  cdlemg2m  30072
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-sep 4142  ax-nul 4150  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-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  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-fv 5229  df-ov 5823  df-lat 14148  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-lhyp 29456
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