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Theorem atcvrneN 33383
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 33314 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
213ad2ant1 1009 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  K  e.  AtLat )
3 simp21 1021 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  P  e.  A )
4 eqid 2451 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
5 atcvrne.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atn0 33262 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  ( 0. `  K
) )
72, 3, 6syl2anc 661 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  P  =/=  ( 0. `  K
) )
8 simp1 988 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  K  e.  HL )
9 eqid 2451 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
109, 5atbase 33243 . . . . 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 1022 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  Q  e.  A )
13 simp23 1023 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  R  e.  A )
14 simp3 990 . . . 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 33381 . . . 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 1232 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  P C
( Q  .\/  R
) )  ->  ( P  =  ( 0. `  K )  <->  Q  =  R ) )
1918necon3bid 2706 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   joincjn 15225   0.cp0 15318    <o ccvr 33216   Atomscatm 33217   AtLatcal 33218   HLchlt 33304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305
This theorem is referenced by:  atleneN  33387
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