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Theorem lnnat 30286
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 961 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
2 simpl2 962 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
3 eqid 2438 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2438 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5 lnnat.a . . . . . . 7  |-  A  =  ( Atoms `  K )
63, 4, 5atcvr0 30148 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( 0. `  K
) (  <o  `  K
) P )
71, 2, 6syl2anc 644 . . . . 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 30276 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P (  <o  `  K )
( P  .\/  Q
) ) )
109biimpa 472 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P (  <o  `  K ) ( P  .\/  Q ) )
11 hlop 30222 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
12 eqid 2438 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 3op0cl 30044 . . . . . . 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 30149 . . . . . . 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 30223 . . . . . . . 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 963 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
2012, 5atbase 30149 . . . . . . . 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 14481 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2318, 16, 21, 22syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
2412, 4cvrntr 30284 . . . . . 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 1187 . . . . 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 662 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  -.  ( 0. `  K ) ( 
<o  `  K ) ( P  .\/  Q ) )
27 simpll1 997 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  Q )  e.  A )  ->  K  e.  HL )
283, 4, 5atcvr0 30148 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  .\/  Q )  e.  A )  -> 
( 0. `  K
) (  <o  `  K
) ( P  .\/  Q ) )
2927, 28sylancom 650 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  Q )  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) ( P  .\/  Q ) )
3026, 29mtand 642 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  -.  ( P  .\/  Q )  e.  A )
3130ex 425 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  ( P  .\/  Q )  e.  A ) )
328, 5hlatjidm 30228 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
33323adant3 978 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  =  P )
34 simp2 959 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  A )
3533, 34eqeltrd 2512 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  e.  A )
36 oveq2 6091 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
3736eleq1d 2504 . . . 4  |-  ( P  =  Q  ->  (
( P  .\/  P
)  e.  A  <->  ( P  .\/  Q )  e.  A
) )
3835, 37syl5ibcom 213 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  ( P  .\/  Q )  e.  A ) )
3938necon3bd 2640 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( -.  ( P 
.\/  Q )  e.  A  ->  P  =/=  Q ) )
4031, 39impbid 185 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   joincjn 14403   0.cp0 14468   Latclat 14476   OPcops 30032    <o ccvr 30122   Atomscatm 30123   HLchlt 30210
This theorem is referenced by:  2atjlej  30338  cdleme11h  31125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211
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