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Theorem llnle 29778
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 730 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  K  e.  HL )
2 simplr 731 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  X  e.  B )
3 simprl 732 . . 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 29696 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
91, 2, 3, 8syl3anc 1183 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. p  e.  A  p  .<_  X )
10 simp1ll 1019 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  K  e.  HL )
114, 7atbase 29550 . . . . . . 7  |-  ( p  e.  A  ->  p  e.  B )
12113ad2ant2 978 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  e.  B
)
13 simp1lr 1020 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  X  e.  B
)
14 simp3 958 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  .<_  X )
15 simp2 957 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  e.  A
)
16 simp1rr 1022 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  -.  X  e.  A )
17 nelne2 2619 . . . . . . . 8  |-  ( ( p  e.  A  /\  -.  X  e.  A
)  ->  p  =/=  X )
1815, 16, 17syl2anc 642 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  =/=  X
)
19 eqid 2366 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
205, 19pltval 14304 . . . . . . . 8  |-  ( ( K  e.  HL  /\  p  e.  A  /\  X  e.  B )  ->  ( p ( lt
`  K ) X  <-> 
( p  .<_  X  /\  p  =/=  X ) ) )
2110, 15, 13, 20syl3anc 1183 . . . . . . 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 888 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p ( lt
`  K ) X )
23 eqid 2366 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
24 eqid 2366 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
254, 5, 19, 23, 24, 7hlrelat3 29672 . . . . . 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 1186 . . . . 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 1019 . . . . . . . . . . 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 989 . . . . . . . . . . . 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 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 ) )  -> 
q  e.  A )
304, 23, 7hlatjcl 29627 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p ( join `  K ) q )  e.  B )
3127, 28, 29, 30syl3anc 1183 . . . . . . . . . . 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 984 . . . . . . . . . . 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 29768 . . . . . . . . . . 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 1186 . . . . . . . . . 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 985 . . . . . . . . . 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 4128 . . . . . . . . . . 11  |-  ( y  =  ( p (
join `  K )
q )  ->  (
y  .<_  X  <->  ( p
( join `  K )
q )  .<_  X ) )
3837rspcev 2969 . . . . . . . . . 10  |-  ( ( ( p ( join `  K ) q )  e.  N  /\  (
p ( join `  K
) q )  .<_  X )  ->  E. y  e.  N  y  .<_  X )
3935, 36, 38syl2anc 642 . . . . . . . . 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 1151 . . . . . . . 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 1169 . . . . . . 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 1146 . . . . . 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 2751 . . . . 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 14 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  E. y  e.  N  y  .<_  X )
45443exp 1151 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  -> 
( p  e.  A  ->  ( p  .<_  X  ->  E. y  e.  N  y  .<_  X ) ) )
4645rexlimdv 2751 . 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 14 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 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   ltcplt 14285   joincjn 14288   0.cp0 14353    <o ccvr 29523   Atomscatm 29524   HLchlt 29611   LLinesclln 29751
This theorem is referenced by:  llnmlplnN  29799  lplnle  29800  llncvrlpln  29818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758
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