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Theorem dalemqnet 33262
Description: Lemma for dath 33346. 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 33234 . . 3  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 33248 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51, 3dalemteb 33253 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
61, 3dalemueb 33254 . . 3  |-  ( ph  ->  U  e.  ( Base `  K ) )
7 simp322 1165 . . . 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 200 . . 3  |-  ( ph  ->  -.  C  .<_  ( T 
.\/  U ) )
9 eqid 2462 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
11 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
129, 10, 11latnlej2l 16367 . . 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 1289 . 2  |-  ( ph  ->  -.  C  .<_  T )
141dalemclqjt 33245 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
15 oveq1 6322 . . . . . 6  |-  ( Q  =  T  ->  ( Q  .\/  T )  =  ( T  .\/  T
) )
1615breq2d 4428 . . . . 5  |-  ( Q  =  T  ->  ( C  .<_  ( Q  .\/  T )  <->  C  .<_  ( T 
.\/  T ) ) )
1714, 16syl5ibcom 228 . . . 4  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  ( T 
.\/  T ) ) )
181dalemkehl 33233 . . . . . 6  |-  ( ph  ->  K  e.  HL )
191dalemtea 33240 . . . . . 6  |-  ( ph  ->  T  e.  A )
2011, 3hlatjidm 32979 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A )  ->  ( T  .\/  T
)  =  T )
2118, 19, 20syl2anc 671 . . . . 5  |-  ( ph  ->  ( T  .\/  T
)  =  T )
2221breq2d 4428 . . . 4  |-  ( ph  ->  ( C  .<_  ( T 
.\/  T )  <->  C  .<_  T ) )
2317, 22sylibd 222 . . 3  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  T ) )
2423necon3bd 2650 . 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   Basecbs 15170   lecple 15246   joincjn 16238   Latclat 16340   Atomscatm 32874   HLchlt 32961   LPlanesclpl 33102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-preset 16222  df-poset 16240  df-lub 16269  df-glb 16270  df-join 16271  df-meet 16272  df-lat 16341  df-ats 32878  df-atl 32909  df-cvlat 32933  df-hlat 32962
This theorem is referenced by:  dalemcea  33270  dalem2  33271  dalemdnee  33276
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