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Theorem lnnat 29909
Description: A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)
Hypotheses
Ref Expression
lnnat.j  |-  .\/  =  ( join `  K )
lnnat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
lnnat  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  -.  ( P  .\/  Q
)  e.  A ) )

Proof of Theorem lnnat
StepHypRef Expression
1 simpl1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
2 simpl2 961 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
3 eqid 2404 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2404 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5 lnnat.a . . . . . . 7  |-  A  =  ( Atoms `  K )
63, 4, 5atcvr0 29771 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( 0. `  K
) (  <o  `  K
) P )
71, 2, 6syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( 0. `  K ) (  <o  `  K ) P )
8 lnnat.j . . . . . . 7  |-  .\/  =  ( join `  K )
98, 4, 5atcvr1 29899 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P (  <o  `  K )
( P  .\/  Q
) ) )
109biimpa 471 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P (  <o  `  K ) ( P  .\/  Q ) )
11 hlop 29845 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
12 eqid 2404 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 3op0cl 29667 . . . . . . 7  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
141, 11, 133syl 19 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( 0. `  K )  e.  (
Base `  K )
)
1512, 5atbase 29772 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
162, 15syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  ( Base `  K )
)
17 hllat 29846 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
181, 17syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
19 simpl3 962 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
2012, 5atbase 29772 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2119, 20syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  ( Base `  K )
)
2212, 8latjcl 14434 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2318, 16, 21, 22syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
2412, 4cvrntr 29907 . . . . . 6  |-  ( ( K  e.  HL  /\  ( ( 0. `  K )  e.  (
Base `  K )  /\  P  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( ( 0.
`  K ) ( 
<o  `  K ) P  /\  P (  <o  `  K ) ( P 
.\/  Q ) )  ->  -.  ( 0. `  K ) (  <o  `  K ) ( P 
.\/  Q ) ) )
251, 14, 16, 23, 24syl13anc 1186 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( (
( 0. `  K
) (  <o  `  K
) P  /\  P
(  <o  `  K )
( P  .\/  Q
) )  ->  -.  ( 0. `  K ) (  <o  `  K )
( P  .\/  Q
) ) )
267, 10, 25mp2and 661 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  -.  ( 0. `  K ) ( 
<o  `  K ) ( P  .\/  Q ) )
27 simpll1 996 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  Q )  e.  A )  ->  K  e.  HL )
283, 4, 5atcvr0 29771 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  .\/  Q )  e.  A )  -> 
( 0. `  K
) (  <o  `  K
) ( P  .\/  Q ) )
2927, 28sylancom 649 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  Q )  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) ( P  .\/  Q ) )
3026, 29mtand 641 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  -.  ( P  .\/  Q )  e.  A )
3130ex 424 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  ( P  .\/  Q )  e.  A ) )
328, 5hlatjidm 29851 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
33323adant3 977 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  =  P )
34 simp2 958 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  A )
3533, 34eqeltrd 2478 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  e.  A )
36 oveq2 6048 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
3736eleq1d 2470 . . . 4  |-  ( P  =  Q  ->  (
( P  .\/  P
)  e.  A  <->  ( P  .\/  Q )  e.  A
) )
3835, 37syl5ibcom 212 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  ( P  .\/  Q )  e.  A ) )
3938necon3bd 2604 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( -.  ( P 
.\/  Q )  e.  A  ->  P  =/=  Q ) )
4031, 39impbid 184 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  -.  ( P  .\/  Q
)  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   joincjn 14356   0.cp0 14421   Latclat 14429   OPcops 29655    <o ccvr 29745   Atomscatm 29746   HLchlt 29833
This theorem is referenced by:  2atjlej  29961  cdleme11h  30748
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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