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Theorem dalemqnet 34741
Description: Lemma for dath 34825. 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
dalemqnet  |-  ( ph  ->  Q  =/=  T )

Proof of Theorem dalemqnet
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 34713 . . 3  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 34727 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51, 3dalemteb 34732 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
61, 3dalemueb 34733 . . 3  |-  ( ph  ->  U  e.  ( Base `  K ) )
7 simp322 1147 . . . 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  .<_  ( T  .\/  U ) )
81, 7sylbi 195 . . 3  |-  ( ph  ->  -.  C  .<_  ( T 
.\/  U ) )
9 eqid 2467 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
11 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
129, 10, 11latnlej2l 15571 . . 3  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  /\  -.  C  .<_  ( T  .\/  U ) )  ->  -.  C  .<_  T )
132, 4, 5, 6, 8, 12syl131anc 1241 . 2  |-  ( ph  ->  -.  C  .<_  T )
141dalemclqjt 34724 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
15 oveq1 6301 . . . . . 6  |-  ( Q  =  T  ->  ( Q  .\/  T )  =  ( T  .\/  T
) )
1615breq2d 4464 . . . . 5  |-  ( Q  =  T  ->  ( C  .<_  ( Q  .\/  T )  <->  C  .<_  ( T 
.\/  T ) ) )
1714, 16syl5ibcom 220 . . . 4  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  ( T 
.\/  T ) ) )
181dalemkehl 34712 . . . . . 6  |-  ( ph  ->  K  e.  HL )
191dalemtea 34719 . . . . . 6  |-  ( ph  ->  T  e.  A )
2011, 3hlatjidm 34458 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A )  ->  ( T  .\/  T
)  =  T )
2118, 19, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( T  .\/  T
)  =  T )
2221breq2d 4464 . . . 4  |-  ( ph  ->  ( C  .<_  ( T 
.\/  T )  <->  C  .<_  T ) )
2317, 22sylibd 214 . . 3  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  T ) )
2423necon3bd 2679 . 2  |-  ( ph  ->  ( -.  C  .<_  T  ->  Q  =/=  T
) )
2513, 24mpd 15 1  |-  ( ph  ->  Q  =/=  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   Basecbs 14502   lecple 14574   joincjn 15443   Latclat 15544   Atomscatm 34353   HLchlt 34440   LPlanesclpl 34581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-poset 15445  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-lat 15545  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441
This theorem is referenced by:  dalemcea  34749  dalem2  34750  dalemdnee  34755
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