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Theorem dalem48 30531
Description: Lemma for dath 30547. Analog of dalem45 30528 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 30435 . . 3  |-  ( ph  ->  K  e.  Lat )
32adantr 451 . 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 30500 . . 3  |-  ( ps 
->  c  e.  ( Base `  K ) )
76adantl 452 . 2  |-  ( (
ph  /\  ps )  ->  c  e.  ( Base `  K ) )
8 dalem.j . . . 4  |-  .\/  =  ( join `  K )
91, 8, 5dalempjqeb 30456 . . 3  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
109adantr 451 . 2  |-  ( (
ph  /\  ps )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
111, 5dalemreb 30452 . . 3  |-  ( ph  ->  R  e.  ( Base `  K ) )
1211adantr 451 . 2  |-  ( (
ph  /\  ps )  ->  R  e.  ( Base `  K ) )
134dalem-ccly 30496 . . . 4  |-  ( ps 
->  -.  c  .<_  Y )
14 dalem44.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1514breq2i 4047 . . . 4  |-  ( c 
.<_  Y  <->  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
1613, 15sylnib 295 . . 3  |-  ( ps 
->  -.  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
1716adantl 452 . 2  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
18 eqid 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
19 dalem.l . . 3  |-  .<_  =  ( le `  K )
2018, 19, 8latnlej2l 14194 . 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 1195 1  |-  ( (
ph  /\  ps )  ->  -.  c  .<_  ( P 
.\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   HLchlt 30162   LPlanesclpl 30303
This theorem is referenced by:  dalem49  30532  dalem51  30534  dalem52  30535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-join 14126  df-lat 14168  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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