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Theorem dalemqnet 35078
Description: Lemma for dath 35162. 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 35050 . . 3  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 35064 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51, 3dalemteb 35069 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
61, 3dalemueb 35070 . . 3  |-  ( ph  ->  U  e.  ( Base `  K ) )
7 simp322 1146 . . . 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 2441 . . . 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 1240 . 2  |-  ( ph  ->  -.  C  .<_  T )
141dalemclqjt 35061 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
15 oveq1 6284 . . . . . 6  |-  ( Q  =  T  ->  ( Q  .\/  T )  =  ( T  .\/  T
) )
1615breq2d 4445 . . . . 5  |-  ( Q  =  T  ->  ( C  .<_  ( Q  .\/  T )  <->  C  .<_  ( T 
.\/  T ) ) )
1714, 16syl5ibcom 220 . . . 4  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  ( T 
.\/  T ) ) )
181dalemkehl 35049 . . . . . 6  |-  ( ph  ->  K  e.  HL )
191dalemtea 35056 . . . . . 6  |-  ( ph  ->  T  e.  A )
2011, 3hlatjidm 34795 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A )  ->  ( T  .\/  T
)  =  T )
2118, 19, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( T  .\/  T
)  =  T )
2221breq2d 4445 . . . 4  |-  ( ph  ->  ( C  .<_  ( T 
.\/  T )  <->  C  .<_  T ) )
2317, 22sylibd 214 . . 3  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  T ) )
2423necon3bd 2653 . 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   lecple 14576   joincjn 15442   Latclat 15544   Atomscatm 34690   HLchlt 34777   LPlanesclpl 34918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-preset 15426  df-poset 15444  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-lat 15545  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778
This theorem is referenced by:  dalemcea  35086  dalem2  35087  dalemdnee  35092
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