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Theorem dalem48 30454
Description: Lemma for dath 30470. Analog of dalem45 30451 for  P Q. (Contributed by NM, 16-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem44.m  |-  ./\  =  ( meet `  K )
dalem44.o  |-  O  =  ( LPlanes `  K )
dalem44.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem44.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem44.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem44.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem44.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
Assertion
Ref Expression
dalem48  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( P 
.\/  Q ) )

Proof of Theorem dalem48
StepHypRef Expression
1 dalem.ph . . . 4  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
21dalemkelat 30358 . . 3  |-  ( ph  ->  K  e.  Lat )
32adantr 452 . 2  |-  ( (
ph  /\  ps )  ->  K  e.  Lat )
4 dalem.ps . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
5 dalem.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5dalemcceb 30423 . . 3  |-  ( ps 
->  c  e.  ( Base `  K ) )
76adantl 453 . 2  |-  ( (
ph  /\  ps )  ->  c  e.  ( Base `  K ) )
8 dalem.j . . . 4  |-  .\/  =  ( join `  K )
91, 8, 5dalempjqeb 30379 . . 3  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
109adantr 452 . 2  |-  ( (
ph  /\  ps )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
111, 5dalemreb 30375 . . 3  |-  ( ph  ->  R  e.  ( Base `  K ) )
1211adantr 452 . 2  |-  ( (
ph  /\  ps )  ->  R  e.  ( Base `  K ) )
134dalem-ccly 30419 . . . 4  |-  ( ps 
->  -.  c  .<_  Y )
14 dalem44.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1514breq2i 4212 . . . 4  |-  ( c 
.<_  Y  <->  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
1613, 15sylnib 296 . . 3  |-  ( ps 
->  -.  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
1716adantl 453 . 2  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
18 eqid 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
19 dalem.l . . 3  |-  .<_  =  ( le `  K )
2018, 19, 8latnlej2l 14493 . 2  |-  ( ( K  e.  Lat  /\  ( c  e.  (
Base `  K )  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  /\  -.  c  .<_  ( ( P 
.\/  Q )  .\/  R ) )  ->  -.  c  .<_  ( P  .\/  Q ) )
213, 7, 10, 12, 17, 20syl131anc 1197 1  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( P 
.\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   Atomscatm 29998   HLchlt 30085   LPlanesclpl 30226
This theorem is referenced by:  dalem49  30455  dalem51  30457  dalem52  30458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-lub 14423  df-join 14425  df-lat 14467  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086
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