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Theorem dalemqnet 35789
Description: Lemma for dath 35873. 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 35761 . . 3  |-  ( ph  ->  K  e.  Lat )
3 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
41, 3dalemceb 35775 . . 3  |-  ( ph  ->  C  e.  ( Base `  K ) )
51, 3dalemteb 35780 . . 3  |-  ( ph  ->  T  e.  ( Base `  K ) )
61, 3dalemueb 35781 . . 3  |-  ( ph  ->  U  e.  ( Base `  K ) )
7 simp322 1145 . . . 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 2382 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
10 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
11 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
129, 10, 11latnlej2l 15819 . . 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 1239 . 2  |-  ( ph  ->  -.  C  .<_  T )
141dalemclqjt 35772 . . . . 5  |-  ( ph  ->  C  .<_  ( Q  .\/  T ) )
15 oveq1 6203 . . . . . 6  |-  ( Q  =  T  ->  ( Q  .\/  T )  =  ( T  .\/  T
) )
1615breq2d 4379 . . . . 5  |-  ( Q  =  T  ->  ( C  .<_  ( Q  .\/  T )  <->  C  .<_  ( T 
.\/  T ) ) )
1714, 16syl5ibcom 220 . . . 4  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  ( T 
.\/  T ) ) )
181dalemkehl 35760 . . . . . 6  |-  ( ph  ->  K  e.  HL )
191dalemtea 35767 . . . . . 6  |-  ( ph  ->  T  e.  A )
2011, 3hlatjidm 35506 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A )  ->  ( T  .\/  T
)  =  T )
2118, 19, 20syl2anc 659 . . . . 5  |-  ( ph  ->  ( T  .\/  T
)  =  T )
2221breq2d 4379 . . . 4  |-  ( ph  ->  ( C  .<_  ( T 
.\/  T )  <->  C  .<_  T ) )
2317, 22sylibd 214 . . 3  |-  ( ph  ->  ( Q  =  T  ->  C  .<_  T ) )
2423necon3bd 2594 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   lecple 14709   joincjn 15690   Latclat 15792   Atomscatm 35401   HLchlt 35488   LPlanesclpl 35629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-preset 15674  df-poset 15692  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-lat 15793  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489
This theorem is referenced by:  dalemcea  35797  dalem2  35798  dalemdnee  35803
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