Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme0aa Unicode version

Theorem cdleme0aa 30375
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 29529 . . . 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 29455 . . . . 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 29455 . . . . 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 14399 . . . 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 30163 . . . 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 14400 . . 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 2464 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 1717   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   meetcmee 14322   Latclat 14394   Atomscatm 29429   HLchlt 29516   LHypclh 30149
This theorem is referenced by:  cdleme1b  30391  cdleme5  30405  cdleme9  30418  cdleme11g  30430  cdleme11  30435  cdleme35fnpq  30614  cdleme42e  30644  cdlemeg46frv  30690  cdlemg2fv2  30765  cdlemg2m  30769
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-lat 14395  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-lhyp 30153
  Copyright terms: Public domain W3C validator