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

Theorem dalempnes 34322
Description: Lemma for dath 34407. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)
Hypotheses
Ref Expression
dalema.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 ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalempnes.o  |-  O  =  ( LPlanes `  K )
dalempnes.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalempnes  |-  ( ph  ->  P  =/=  S )

Proof of Theorem dalempnes
StepHypRef Expression
1 dalema.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 34295 . . 3  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 34309 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51, 3dalemseb 34313 . . 3  |-  ( ph  ->  S  e.  ( Base `  K ) )
61, 3dalemteb 34314 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
7 simp321 1141 . . . 4  |-  ( ( ( ( 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 ) ) ) )  ->  -.  C  .<_  ( S  .\/  T ) )
81, 7sylbi 195 . . 3  |-  ( ph  ->  -.  C  .<_  ( S 
.\/  T ) )
9 eqid 2460 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
11 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
129, 10, 11latnlej2l 15548 . . 3  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) )  /\  -.  C  .<_  ( S  .\/  T ) )  ->  -.  C  .<_  S )
132, 4, 5, 6, 8, 12syl131anc 1236 . 2  |-  ( ph  ->  -.  C  .<_  S )
141dalemclpjs 34305 . . . . 5  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
15 oveq1 6282 . . . . . 6  |-  ( P  =  S  ->  ( P  .\/  S )  =  ( S  .\/  S
) )
1615breq2d 4452 . . . . 5  |-  ( P  =  S  ->  ( C  .<_  ( P  .\/  S )  <->  C  .<_  ( S 
.\/  S ) ) )
1714, 16syl5ibcom 220 . . . 4  |-  ( ph  ->  ( P  =  S  ->  C  .<_  ( S 
.\/  S ) ) )
181dalemkehl 34294 . . . . . 6  |-  ( ph  ->  K  e.  HL )
191dalemsea 34300 . . . . . 6  |-  ( ph  ->  S  e.  A )
2011, 3hlatjidm 34040 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A )  ->  ( S  .\/  S
)  =  S )
2118, 19, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( S  .\/  S
)  =  S )
2221breq2d 4452 . . . 4  |-  ( ph  ->  ( C  .<_  ( S 
.\/  S )  <->  C  .<_  S ) )
2317, 22sylibd 214 . . 3  |-  ( ph  ->  ( P  =  S  ->  C  .<_  S ) )
2423necon3bd 2672 . 2  |-  ( ph  ->  ( -.  C  .<_  S  ->  P  =/=  S
) )
2513, 24mpd 15 1  |-  ( ph  ->  P  =/=  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   Latclat 15521   Atomscatm 33935   HLchlt 34022   LPlanesclpl 34163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-lat 15522  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023
This theorem is referenced by:  dalempjsen  34324  dalem24  34368
  Copyright terms: Public domain W3C validator