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Theorem cdleme0aa 30696
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 29850 . . . 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 29776 . . . . 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 29776 . . . . 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 14438 . . . 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 30484 . . . 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 14439 . . 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 2492 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 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   Latclat 14433   Atomscatm 29750   HLchlt 29837   LHypclh 30470
This theorem is referenced by:  cdleme1b  30712  cdleme5  30726  cdleme9  30739  cdleme11g  30751  cdleme11  30756  cdleme35fnpq  30935  cdleme42e  30965  cdlemeg46frv  31011  cdlemg2fv2  31086  cdlemg2m  31090
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-lat 14434  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-lhyp 30474
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