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Theorem llnle 34804
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 790 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) -> K e. HL )
2 simplr 792 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) -> X e. B )
3 simprl 794 . . 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 34722 . . 3 |- ( ( K e. HL /\ X e. B /\ X =/= .0. ) -> E. p e. A p .<_ X )
91, 2, 3, 8syl3anc 1326 . 2 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) -> E. p e. A p .<_ X )
10 simp1ll 1124 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) /\ p e. A /\ p .<_ X ) -> K e. HL )
114, 7atbase 34576 . . . . . . 7 |- ( p e. A -> p e. B )
12113ad2ant2 1083 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) /\ p e. A /\ p .<_ X ) -> p e. B )
13 simp1lr 1125 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) /\ p e. A /\ p .<_ X ) -> X e. B )
14 simp3 1063 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) /\ p e. A /\ p .<_ X ) -> p .<_ X )
15 simp2 1062 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) /\ p e. A /\ p .<_ X ) -> p e. A )
16 simp1rr 1127 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) /\ p e. A /\ p .<_ X ) -> -. X e. A )
17 nelne2 2891 . . . . . . . 8 |- ( ( p e. A /\ -. X e. A ) -> p =/= X )
1815, 16, 17syl2anc 693 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) /\ p e. A /\ p .<_ X ) -> p =/= X )
19 eqid 2622 . . . . . . . . 9 |- ( lt ` K ) = ( lt ` K )
205, 19pltval 16960 . . . . . . . 8 |- ( ( K e. HL /\ p e. A /\ X e. B ) -> ( p ( lt ` K ) X <-> ( p .<_ X /\ p =/= X ) ) )
2110, 15, 13, 20syl3anc 1326 . . . . . . 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 957 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) /\ p e. A /\ p .<_ X ) -> p ( lt ` K ) X )
23 eqid 2622 . . . . . . 7 |- ( join ` K ) = ( join ` K )
24 eqid 2622 . . . . . . 7 |- ( <o ` K ) = ( <o ` K )
254, 5, 19, 23, 24, 7hlrelat3 34698 . . . . . 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 1329 . . . . 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 1124 . . . . . . . . . . 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 1094 . . . . . . . . . . . 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 1096 . . . . . . . . . . . 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 34653 . . . . . . . . . . . 12 |- ( ( K e. HL /\ p e. A /\ q e. A ) -> ( p ( join ` K ) q ) e. B )
3127, 28, 29, 30syl3anc 1326 . . . . . . . . . . 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 1089 . . . . . . . . . . 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 34794 . . . . . . . . . . 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 1329 . . . . . . . . . 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 1090 . . . . . . . . . 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 4656 . . . . . . . . . . 11 |- ( y = ( p ( join ` K ) q ) -> ( y .<_ X <-> ( p ( join ` K ) q ) .<_ X ) )
3837rspcev 3309 . . . . . . . . . 10 |- ( ( ( p ( join ` K ) q ) e. N /\ ( p ( join ` K ) q ) .<_ X ) -> E. y e. N y .<_ X )
3935, 36, 38syl2anc 693 . . . . . . . . 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 1264 . . . . . . . 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 1284 . . . . . . 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 1256 . . . . . 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 3030 . . . . 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 1264 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) -> ( p e. A -> ( p .<_ X -> E. y e. N y .<_ X ) ) )
4645rexlimdv 3030 . 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   ltcplt 16941   joincjn 16944   0.cp0 17037   <o ccvr 34549   Atomscatm 34550   HLchlt 34637   LLinesclln 34777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784
This theorem is referenced by:  llnmlplnN  34825  lplnle  34826  llncvrlpln  34844
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