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Theorem llnle 32515
Description: Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)
Hypotheses
Ref Expression
llnle.b  |-  B  =  ( Base `  K
)
llnle.l  |-  .<_  =  ( le `  K )
llnle.z  |-  .0.  =  ( 0. `  K )
llnle.a  |-  A  =  ( Atoms `  K )
llnle.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llnle  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. y  e.  N  y  .<_  X )
Distinct variable groups:    y, K    y, 
.<_    y, N    y, X
Allowed substitution hints:    A( y)    B( y)    .0. ( y)

Proof of Theorem llnle
Dummy variables  q  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 752 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  K  e.  HL )
2 simplr 754 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  X  e.  B )
3 simprl 756 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  X  =/=  .0.  )
4 llnle.b . . . 4  |-  B  =  ( Base `  K
)
5 llnle.l . . . 4  |-  .<_  =  ( le `  K )
6 llnle.z . . . 4  |-  .0.  =  ( 0. `  K )
7 llnle.a . . . 4  |-  A  =  ( Atoms `  K )
84, 5, 6, 7atle 32433 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
91, 2, 3, 8syl3anc 1230 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. p  e.  A  p  .<_  X )
10 simp1ll 1060 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  K  e.  HL )
114, 7atbase 32287 . . . . . . 7  |-  ( p  e.  A  ->  p  e.  B )
12113ad2ant2 1019 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  e.  B
)
13 simp1lr 1061 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  X  e.  B
)
14 simp3 999 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  .<_  X )
15 simp2 998 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  e.  A
)
16 simp1rr 1063 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  -.  X  e.  A )
17 nelne2 2733 . . . . . . . 8  |-  ( ( p  e.  A  /\  -.  X  e.  A
)  ->  p  =/=  X )
1815, 16, 17syl2anc 659 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  =/=  X
)
19 eqid 2402 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
205, 19pltval 15912 . . . . . . . 8  |-  ( ( K  e.  HL  /\  p  e.  A  /\  X  e.  B )  ->  ( p ( lt
`  K ) X  <-> 
( p  .<_  X  /\  p  =/=  X ) ) )
2110, 15, 13, 20syl3anc 1230 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  ( p ( lt `  K ) X  <->  ( p  .<_  X  /\  p  =/=  X
) ) )
2214, 18, 21mpbir2and 923 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p ( lt
`  K ) X )
23 eqid 2402 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
24 eqid 2402 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
254, 5, 19, 23, 24, 7hlrelat3 32409 . . . . . 6  |-  ( ( ( K  e.  HL  /\  p  e.  B  /\  X  e.  B )  /\  p ( lt `  K ) X )  ->  E. q  e.  A  ( p (  <o  `  K ) ( p ( join `  K
) q )  /\  ( p ( join `  K ) q ) 
.<_  X ) )
2610, 12, 13, 22, 25syl31anc 1233 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  E. q  e.  A  ( p (  <o  `  K ) ( p ( join `  K
) q )  /\  ( p ( join `  K ) q ) 
.<_  X ) )
27 simp1ll 1060 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  ( p  e.  A  /\  p  .<_  X  /\  q  e.  A )  /\  (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X ) )  ->  K  e.  HL )
28 simp21 1030 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  ( p  e.  A  /\  p  .<_  X  /\  q  e.  A )  /\  (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X ) )  ->  p  e.  A )
29 simp23 1032 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  ( p  e.  A  /\  p  .<_  X  /\  q  e.  A )  /\  (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X ) )  -> 
q  e.  A )
304, 23, 7hlatjcl 32364 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p ( join `  K ) q )  e.  B )
3127, 28, 29, 30syl3anc 1230 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  ( p  e.  A  /\  p  .<_  X  /\  q  e.  A )  /\  (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X ) )  -> 
( p ( join `  K ) q )  e.  B )
32 simp3l 1025 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  ( p  e.  A  /\  p  .<_  X  /\  q  e.  A )  /\  (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X ) )  ->  p (  <o  `  K
) ( p (
join `  K )
q ) )
33 llnle.n . . . . . . . . . . . 12  |-  N  =  ( LLines `  K )
344, 24, 7, 33llni 32505 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( p ( join `  K ) q )  e.  B  /\  p  e.  A )  /\  p
(  <o  `  K )
( p ( join `  K ) q ) )  ->  ( p
( join `  K )
q )  e.  N
)
3527, 31, 28, 32, 34syl31anc 1233 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  ( p  e.  A  /\  p  .<_  X  /\  q  e.  A )  /\  (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X ) )  -> 
( p ( join `  K ) q )  e.  N )
36 simp3r 1026 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  ( p  e.  A  /\  p  .<_  X  /\  q  e.  A )  /\  (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X ) )  -> 
( p ( join `  K ) q ) 
.<_  X )
37 breq1 4397 . . . . . . . . . . 11  |-  ( y  =  ( p (
join `  K )
q )  ->  (
y  .<_  X  <->  ( p
( join `  K )
q )  .<_  X ) )
3837rspcev 3159 . . . . . . . . . 10  |-  ( ( ( p ( join `  K ) q )  e.  N  /\  (
p ( join `  K
) q )  .<_  X )  ->  E. y  e.  N  y  .<_  X )
3935, 36, 38syl2anc 659 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  ( p  e.  A  /\  p  .<_  X  /\  q  e.  A )  /\  (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X ) )  ->  E. y  e.  N  y  .<_  X )
40393exp 1196 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  -> 
( ( p  e.  A  /\  p  .<_  X  /\  q  e.  A
)  ->  ( (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X )  ->  E. y  e.  N  y  .<_  X ) ) )
41403expd 1214 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  -> 
( p  e.  A  ->  ( p  .<_  X  -> 
( q  e.  A  ->  ( ( p ( 
<o  `  K ) ( p ( join `  K
) q )  /\  ( p ( join `  K ) q ) 
.<_  X )  ->  E. y  e.  N  y  .<_  X ) ) ) ) )
42413imp 1191 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  ( q  e.  A  ->  ( (
p (  <o  `  K
) ( p (
join `  K )
q )  /\  (
p ( join `  K
) q )  .<_  X )  ->  E. y  e.  N  y  .<_  X ) ) )
4342rexlimdv 2893 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  ( E. q  e.  A  ( p
(  <o  `  K )
( p ( join `  K ) q )  /\  ( p (
join `  K )
q )  .<_  X )  ->  E. y  e.  N  y  .<_  X ) )
4426, 43mpd 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  E. y  e.  N  y  .<_  X )
45443exp 1196 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  -> 
( p  e.  A  ->  ( p  .<_  X  ->  E. y  e.  N  y  .<_  X ) ) )
4645rexlimdv 2893 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  -> 
( E. p  e.  A  p  .<_  X  ->  E. y  e.  N  y  .<_  X ) )
479, 46mpd 15 1  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. y  e.  N  y  .<_  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   ltcplt 15892   joincjn 15895   0.cp0 15989    <o ccvr 32260   Atomscatm 32261   HLchlt 32348   LLinesclln 32488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495
This theorem is referenced by:  llnmlplnN  32536  lplnle  32537  llncvrlpln  32555
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