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Theorem atcvrneN 29544
Description: Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atcvrne.j  |-  .\/  =  ( join `  K )
atcvrne.c  |-  C  =  (  <o  `  K )
atcvrne.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atcvrneN  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  Q  =/=  R )

Proof of Theorem atcvrneN
StepHypRef Expression
1 hlatl 29475 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
213ad2ant1 978 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  K  e.  AtLat )
3 simp21 990 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  P  e.  A )
4 eqid 2387 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
5 atcvrne.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atn0 29423 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  ( 0. `  K
) )
72, 3, 6syl2anc 643 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  P  =/=  ( 0. `  K
) )
8 simp1 957 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  K  e.  HL )
9 eqid 2387 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
109, 5atbase 29404 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
113, 10syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  P  e.  ( Base `  K
) )
12 simp22 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  Q  e.  A )
13 simp23 992 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  R  e.  A )
14 simp3 959 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  P C ( Q  .\/  R ) )
15 atcvrne.j . . . . 5  |-  .\/  =  ( join `  K )
16 atcvrne.c . . . . 5  |-  C  =  (  <o  `  K )
179, 15, 4, 16, 5atcvrj0 29542 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  ( Base `  K )  /\  Q  e.  A  /\  R  e.  A )  /\  P C ( Q 
.\/  R ) )  ->  ( P  =  ( 0. `  K
)  <->  Q  =  R
) )
188, 11, 12, 13, 14, 17syl131anc 1197 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  ( P  =  ( 0. `  K )  <->  Q  =  R ) )
1918necon3bid 2585 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  ( P  =/=  ( 0. `  K )  <->  Q  =/=  R ) )
207, 19mpbid 202 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  Q  =/=  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   joincjn 14328   0.cp0 14393    <o ccvr 29377   Atomscatm 29378   AtLatcal 29379   HLchlt 29465
This theorem is referenced by:  atleneN  29548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466
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