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Theorem cdleme0aa 30692
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 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  K  e.  HL )
3 hllat 29846 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . 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 29772 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
873ad2ant2 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  P  e.  B )
95, 6atbase 29772 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
1093ad2ant3 980 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  Q  e.  B )
11 cdleme0.j . . . . 5  |-  .\/  =  ( join `  K )
125, 11latjcl 14434 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
134, 8, 10, 12syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  ( P  .\/  Q )  e.  B
)
14 simp1r 982 . . . 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 30480 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
1714, 16syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  W  e.  B )
18 cdleme0.m . . . 4  |-  ./\  =  ( meet `  K )
195, 18latmcl 14435 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  Q
)  ./\  W )  e.  B )
204, 13, 17, 19syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  B )
211, 20syl5eqel 2488 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme1b  30708  cdleme5  30722  cdleme9  30735  cdleme11g  30747  cdleme11  30752  cdleme35fnpq  30931  cdleme42e  30961  cdlemeg46frv  31007  cdlemg2fv2  31082  cdlemg2m  31086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-lat 14430  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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