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Theorem atcvrneN 30164
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 30095 . . . 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 2435 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
5 atcvrne.a . . . 4  |-  A  =  ( Atoms `  K )
64, 5atn0 30043 . . 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 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
109, 5atbase 30024 . . . . 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 30162 . . . 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 2633 . 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   joincjn 14393   0.cp0 14458    <o ccvr 29997   Atomscatm 29998   AtLatcal 29999   HLchlt 30085
This theorem is referenced by:  atleneN  30168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086
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