Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme4a Structured version   Unicode version

Theorem cdleme4a 34191
Description: Part of proof of Lemma E in [Crawley] p. 114 top.  G represents fs(r). Auxiliary lemma derived from cdleme5 34192. We show fs(r)  <_ p  \/ q. (Contributed by NM, 10-Nov-2012.)
Hypotheses
Ref Expression
cdleme4.l  |-  .<_  =  ( le `  K )
cdleme4.j  |-  .\/  =  ( join `  K )
cdleme4.m  |-  ./\  =  ( meet `  K )
cdleme4.a  |-  A  =  ( Atoms `  K )
cdleme4.h  |-  H  =  ( LHyp `  K
)
cdleme4.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme4.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme4.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme4a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  G  .<_  ( P  .\/  Q
) )

Proof of Theorem cdleme4a
StepHypRef Expression
1 cdleme4.g . 2  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
2 simp1l 1012 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  K  e.  HL )
3 hllat 33316 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  K  e.  Lat )
5 simp21 1021 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  P  e.  A )
6 simp22 1022 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  Q  e.  A )
7 eqid 2451 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme4.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdleme4.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 33319 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
12 simp1r 1013 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  W  e.  H )
13 simp3 990 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  S  e.  A )
14 cdleme4.l . . . . . 6  |-  .<_  =  ( le `  K )
15 cdleme4.m . . . . . 6  |-  ./\  =  ( meet `  K )
16 cdleme4.h . . . . . 6  |-  H  =  ( LHyp `  K
)
17 cdleme4.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme4.f . . . . . 6  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
1914, 8, 15, 9, 16, 17, 18, 7cdleme1b 34178 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A ) )  ->  F  e.  ( Base `  K ) )
202, 12, 5, 6, 13, 19syl23anc 1226 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  F  e.  ( Base `  K
) )
21 simp23 1023 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  R  e.  A )
227, 8, 9hlatjcl 33319 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
232, 21, 13, 22syl3anc 1219 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  ( R  .\/  S )  e.  ( Base `  K
) )
247, 16lhpbase 33950 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2512, 24syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  W  e.  ( Base `  K
) )
267, 15latmcl 15326 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  e.  ( Base `  K ) )
274, 23, 25, 26syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  (
( R  .\/  S
)  ./\  W )  e.  ( Base `  K
) )
287, 8latjcl 15325 . . . 4  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  (
( R  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
294, 20, 27, 28syl3anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
307, 14, 15latmle1 15350 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
)  e.  ( Base `  K ) )  -> 
( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )  .<_  ( P 
.\/  Q ) )
314, 11, 29, 30syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )  .<_  ( P 
.\/  Q ) )
321, 31syl5eqbr 4425 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  G  .<_  ( P  .\/  Q
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   meetcmee 15219   Latclat 15319   Atomscatm 33216   HLchlt 33303   LHypclh 33936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-lat 15320  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-lhyp 33940
This theorem is referenced by:  cdleme18c  34245  cdleme41sn3a  34385
  Copyright terms: Public domain W3C validator