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Theorem atcvrneN 35551
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 35482 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
213ad2ant1 1015 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  K  e.  AtLat )
3 simp21 1027 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  P  e.  A )
4 eqid 2454 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
5 atcvrne.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atn0 35430 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  ( 0. `  K
) )
72, 3, 6syl2anc 659 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  P  =/=  ( 0. `  K
) )
8 simp1 994 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  K  e.  HL )
9 eqid 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
109, 5atbase 35411 . . . . 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 1028 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  Q  e.  A )
13 simp23 1029 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  R  e.  A )
14 simp3 996 . . . 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 35549 . . . 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 1239 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  ( P  =  ( 0. `  K )  <->  Q  =  R ) )
1918necon3bid 2712 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  ( P  =/=  ( 0. `  K )  <->  Q  =/=  R ) )
207, 19mpbid 210 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  Q  =/=  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   joincjn 15772   0.cp0 15866    <o ccvr 35384   Atomscatm 35385   AtLatcal 35386   HLchlt 35472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473
This theorem is referenced by:  atleneN  35555
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