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Theorem lnnat 30238
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 2296 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
4 eqid 2296 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
5 lnnat.a . . . . . . 7  |-  A  =  ( Atoms `  K )
63, 4, 5atcvr0 30100 . . . . . 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 30228 . . . . . 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 30174 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OP )
12 eqid 2296 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1312, 3op0cl 29996 . . . . . . 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 30101 . . . . . . 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 30175 . . . . . . . 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 30101 . . . . . . . 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 14172 . . . . . . 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 30236 . . . . . 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 30100 . . . . 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 30180 . . . . . 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 2370 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  P
)  e.  A )
36 oveq2 5882 . . . . 5  |-  ( P  =  Q  ->  ( P  .\/  P )  =  ( P  .\/  Q
) )
3736eleq1d 2362 . . . 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 2496 . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   0.cp0 14159   Latclat 14167   OPcops 29984    <o ccvr 30074   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  2atjlej  30290  cdleme11h  31077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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