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Theorem cdleme0aa 29649
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 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  K  e.  HL )
3 hllat 28803 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . 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 28729 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
873ad2ant2 982 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  P  e.  B )
95, 6atbase 28729 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
1093ad2ant3 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  Q  e.  B )
11 cdleme0.j . . . . 5  |-  .\/  =  ( join `  K )
125, 11latjcl 14119 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
134, 8, 10, 12syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  ( P  .\/  Q )  e.  B
)
14 simp1r 985 . . . 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 29437 . . . 4  |-  ( W  e.  H  ->  W  e.  B )
1714, 16syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  W  e.  B )
18 cdleme0.m . . . 4  |-  ./\  =  ( meet `  K )
195, 18latmcl 14120 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  Q
)  ./\  W )  e.  B )
204, 13, 17, 19syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  B )
211, 20syl5eqel 2342 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4673  (class class class)co 5792   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   Latclat 14114   Atomscatm 28703   HLchlt 28790   LHypclh 29423
This theorem is referenced by:  cdleme1b  29665  cdleme5  29679  cdleme9  29692  cdleme11g  29704  cdleme11  29709  cdleme35fnpq  29888  cdleme42e  29918  cdlemeg46frv  29964  cdlemg2fv2  30039  cdlemg2m  30043
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689  df-ov 5795  df-lat 14115  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-lhyp 29427
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