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Theorem llnle 30004
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 731 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  K  e.  HL )
2 simplr 732 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  X  e.  B )
3 simprl 733 . . 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 29922 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
91, 2, 3, 8syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. p  e.  A  p  .<_  X )
10 simp1ll 1020 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  K  e.  HL )
114, 7atbase 29776 . . . . . . 7  |-  ( p  e.  A  ->  p  e.  B )
12113ad2ant2 979 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  e.  B
)
13 simp1lr 1021 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  X  e.  B
)
14 simp3 959 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  .<_  X )
15 simp2 958 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  e.  A
)
16 simp1rr 1023 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  -.  X  e.  A )
17 nelne2 2661 . . . . . . . 8  |-  ( ( p  e.  A  /\  -.  X  e.  A
)  ->  p  =/=  X )
1815, 16, 17syl2anc 643 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p  =/=  X
)
19 eqid 2408 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
205, 19pltval 14376 . . . . . . . 8  |-  ( ( K  e.  HL  /\  p  e.  A  /\  X  e.  B )  ->  ( p ( lt
`  K ) X  <-> 
( p  .<_  X  /\  p  =/=  X ) ) )
2110, 15, 13, 20syl3anc 1184 . . . . . . 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 889 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A )
)  /\  p  e.  A  /\  p  .<_  X )  ->  p ( lt
`  K ) X )
23 eqid 2408 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
24 eqid 2408 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
254, 5, 19, 23, 24, 7hlrelat3 29898 . . . . . 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 1187 . . . . 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 1020 . . . . . . . . . . 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 990 . . . . . . . . . . . 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 992 . . . . . . . . . . . 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 29853 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  p  e.  A  /\  q  e.  A )  ->  ( p ( join `  K ) q )  e.  B )
3127, 28, 29, 30syl3anc 1184 . . . . . . . . . . 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 985 . . . . . . . . . . 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 29994 . . . . . . . . . . 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 1187 . . . . . . . . . 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 986 . . . . . . . . . 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 4179 . . . . . . . . . . 11  |-  ( y  =  ( p (
join `  K )
q )  ->  (
y  .<_  X  <->  ( p
( join `  K )
q )  .<_  X ) )
3837rspcev 3016 . . . . . . . . . 10  |-  ( ( ( p ( join `  K ) q )  e.  N  /\  (
p ( join `  K
) q )  .<_  X )  ->  E. y  e.  N  y  .<_  X )
3935, 36, 38syl2anc 643 . . . . . . . . 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 1152 . . . . . . . 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 1170 . . . . . . 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 1147 . . . . . 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 2793 . . . . 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 1152 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  -> 
( p  e.  A  ->  ( p  .<_  X  ->  E. y  e.  N  y  .<_  X ) ) )
4645rexlimdv 2793 . 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   ltcplt 14357   joincjn 14360   0.cp0 14425    <o ccvr 29749   Atomscatm 29750   HLchlt 29837   LLinesclln 29977
This theorem is referenced by:  llnmlplnN  30025  lplnle  30026  llncvrlpln  30044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984
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