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Theorem llnle 30389
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 732 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  K  e.  HL )
2 simplr 733 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  X  e.  B )
3 simprl 734 . . 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 30307 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
91, 2, 3, 8syl3anc 1185 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. p  e.  A  p  .<_  X )
10 simp1ll 1021 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  K  e.  HL )
114, 7atbase 30161 . . . . . . 7  |-  ( p  e.  A  ->  p  e.  B )
12113ad2ant2 980 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  e.  B
)
13 simp1lr 1022 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  X  e.  B
)
14 simp3 960 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  .<_  X )
15 simp2 959 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  e.  A
)
16 simp1rr 1024 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  -.  X  e.  A )
17 nelne2 2696 . . . . . . . 8  |-  ( ( p  e.  A  /\  -.  X  e.  A
)  ->  p  =/=  X )
1815, 16, 17syl2anc 644 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  =/=  X
)
19 eqid 2438 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
205, 19pltval 14422 . . . . . . . 8  |-  ( ( K  e.  HL  /\  p  e.  A  /\  X  e.  B )  ->  ( p ( lt
`  K ) X  <-> 
( p  .<_  X  /\  p  =/=  X ) ) )
2110, 15, 13, 20syl3anc 1185 . . . . . . 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 890 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p ( lt
`  K ) X )
23 eqid 2438 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
24 eqid 2438 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
254, 5, 19, 23, 24, 7hlrelat3 30283 . . . . . 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 1188 . . . . 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 1021 . . . . . . . . . . 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 991 . . . . . . . . . . . 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 993 . . . . . . . . . . . 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 30238 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p ( join `  K ) q )  e.  B )
3127, 28, 29, 30syl3anc 1185 . . . . . . . . . . 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 986 . . . . . . . . . . 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 30379 . . . . . . . . . . 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 1188 . . . . . . . . . 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 987 . . . . . . . . . 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 4218 . . . . . . . . . . 11  |-  ( y  =  ( p (
join `  K )
q )  ->  (
y  .<_  X  <->  ( p
( join `  K )
q )  .<_  X ) )
3837rspcev 3054 . . . . . . . . . 10  |-  ( ( ( p ( join `  K ) q )  e.  N  /\  (
p ( join `  K
) q )  .<_  X )  ->  E. y  e.  N  y  .<_  X )
3935, 36, 38syl2anc 644 . . . . . . . . 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 1153 . . . . . . . 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 1171 . . . . . . 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 1148 . . . . . 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 2831 . . . . 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 1153 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  -> 
( p  e.  A  ->  ( p  .<_  X  ->  E. y  e.  N  y  .<_  X ) ) )
4645rexlimdv 2831 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   ltcplt 14403   joincjn 14406   0.cp0 14471    <o ccvr 30134   Atomscatm 30135   HLchlt 30222   LLinesclln 30362
This theorem is referenced by:  llnmlplnN  30410  lplnle  30411  llncvrlpln  30429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369
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