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Theorem lnnat 29616
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 958 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
2 simpl2 959 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
3 eqid 2283 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2283 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5 lnnat.a . . . . . . 7  |-  A  =  ( Atoms `  K )
63, 4, 5atcvr0 29478 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( 0. `  K
) (  <o  `  K
) P )
71, 2, 6syl2anc 642 . . . . 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 29606 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P (  <o  `  K )
( P  .\/  Q
) ) )
109biimpa 470 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P (  <o  `  K ) ( P  .\/  Q ) )
11 hlop 29552 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
12 eqid 2283 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 3op0cl 29374 . . . . . . 7  |-  ( K  e.  OP  ->  ( 0. `  K )  e.  ( Base `  K
) )
141, 11, 133syl 18 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( 0. `  K )  e.  (
Base `  K )
)
1512, 5atbase 29479 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
162, 15syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  ( Base `  K )
)
17 hllat 29553 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
181, 17syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
19 simpl3 960 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
2012, 5atbase 29479 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2119, 20syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  ( Base `  K )
)
2212, 8latjcl 14156 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2318, 16, 21, 22syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
2412, 4cvrntr 29614 . . . . . 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 1184 . . . . 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 660 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  -.  ( 0. `  K ) ( 
<o  `  K ) ( P  .\/  Q ) )
27 simpll1 994 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  Q )  e.  A )  ->  K  e.  HL )
283, 4, 5atcvr0 29478 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  .\/  Q )  e.  A )  -> 
( 0. `  K
) (  <o  `  K
) ( P  .\/  Q ) )
2927, 28sylancom 648 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  /\  ( P  .\/  Q )  e.  A )  ->  ( 0. `  K ) ( 
<o  `  K ) ( P  .\/  Q ) )
3026, 29mtand 640 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  -.  ( P  .\/  Q )  e.  A )
3130ex 423 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  -.  ( P  .\/  Q )  e.  A ) )
328, 5hlatjidm 29558 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P  .\/  P
)  =  P )
33323adant3 975 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  =  P )
34 simp2 956 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  A )
3533, 34eqeltrd 2357 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  e.  A )
36 oveq2 5866 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
3736eleq1d 2349 . . . 4  |-  ( P  =  Q  ->  (
( P  .\/  P
)  e.  A  <->  ( P  .\/  Q )  e.  A
) )
3835, 37syl5ibcom 211 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =  Q  ->  ( P  .\/  Q )  e.  A ) )
3938necon3bd 2483 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( -.  ( P 
.\/  Q )  e.  A  ->  P  =/=  Q ) )
4031, 39impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   0.cp0 14143   Latclat 14151   OPcops 29362    <o ccvr 29452   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  2atjlej  29668  cdleme11h  30455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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