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Theorem lplnle 34826
Description: Any element greater than 0 and not an atom and not a lattice line majorizes a lattice plane. (Contributed by NM, 28-Jun-2012.)
Hypotheses
Ref Expression
lplnle.b |- B = ( Base ` K )
lplnle.l |- .<_ = ( le ` K )
lplnle.z |- .0. = ( 0. ` K )
lplnle.a |- A = ( Atoms ` K )
lplnle.n |- N = ( LLines ` K )
lplnle.p |- P = ( LPlanes ` K )
Assertion
Ref Expression
lplnle |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) -> E. y e. P y .<_ X )
Distinct variable groups:   y, K   y, .<_   y, P   y, X
Allowed substitution hints:   A( y)   B( y)   N( y)   .0. ( y)
Proof of Theorem lplnle
Dummy variables z p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lplnle.b . . . 4 |- B = ( Base ` K )
2 lplnle.l . . . 4 |- .<_ = ( le ` K )
3 lplnle.z . . . 4 |- .0. = ( 0. ` K )
4 lplnle.a . . . 4 |- A = ( Atoms ` K )
5 lplnle.n . . . 4 |- N = ( LLines ` K )
61, 2, 3, 4, 5llnle 34804 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A ) ) -> E. z e. N z .<_ X )
763adantr3 1222 . 2 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) -> E. z e. N z .<_ X )
8 simp1ll 1124 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> K e. HL )
91, 5llnbase 34795 . . . . . . 7 |- ( z e. N -> z e. B )
1093ad2ant2 1083 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> z e. B )
11 simp1lr 1125 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> X e. B )
12 simp3 1063 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> z .<_ X )
13 simp2 1062 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> z e. N )
14 simp1r3 1159 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> -. X e. N )
15 nelne2 2891 . . . . . . . 8 |- ( ( z e. N /\ -. X e. N ) -> z =/= X )
1613, 14, 15syl2anc 693 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> z =/= X )
17 eqid 2622 . . . . . . . . 9 |- ( lt ` K ) = ( lt ` K )
182, 17pltval 16960 . . . . . . . 8 |- ( ( K e. HL /\ z e. N /\ X e. B ) -> ( z ( lt ` K ) X <-> ( z .<_ X /\ z =/= X ) ) )
198, 13, 11, 18syl3anc 1326 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> ( z ( lt ` K ) X <-> ( z .<_ X /\ z =/= X ) ) )
2012, 16, 19mpbir2and 957 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> z ( lt ` K ) X )
21 eqid 2622 . . . . . . 7 |- ( join ` K ) = ( join ` K )
22 eqid 2622 . . . . . . 7 |- ( <o ` K ) = ( <o ` K )
231, 2, 17, 21, 22, 4hlrelat3 34698 . . . . . 6 |- ( ( ( K e. HL /\ z e. B /\ X e. B ) /\ z ( lt ` K ) X ) -> E. p e. A ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) )
248, 10, 11, 20, 23syl31anc 1329 . . . . 5 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> E. p e. A ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) )
25 simp1ll 1124 . . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> K e. HL )
26 hllat 34650 . . . . . . . . . . . . 13 |- ( K e. HL -> K e. Lat )
2725, 26syl 17 . . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> K e. Lat )
28 simp21 1094 . . . . . . . . . . . . 13 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> z e. N )
2928, 9syl 17 . . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> z e. B )
30 simp23 1096 . . . . . . . . . . . . 13 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> p e. A )
311, 4atbase 34576 . . . . . . . . . . . . 13 |- ( p e. A -> p e. B )
3230, 31syl 17 . . . . . . . . . . . 12 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> p e. B )
331, 21latjcl 17051 . . . . . . . . . . . 12 |- ( ( K e. Lat /\ z e. B /\ p e. B ) -> ( z ( join ` K ) p ) e. B )
3427, 29, 32, 33syl3anc 1326 . . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> ( z ( join ` K ) p ) e. B )
35 simp3l 1089 . . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> z ( <o ` K ) ( z ( join ` K ) p ) )
36 lplnle.p . . . . . . . . . . . 12 |- P = ( LPlanes ` K )
371, 22, 5, 36lplni 34818 . . . . . . . . . . 11 |- ( ( ( K e. HL /\ ( z ( join ` K ) p ) e. B /\ z e. N ) /\ z ( <o ` K ) ( z ( join ` K ) p ) ) -> ( z ( join ` K ) p ) e. P )
3825, 34, 28, 35, 37syl31anc 1329 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> ( z ( join ` K ) p ) e. P )
39 simp3r 1090 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> ( z ( join ` K ) p ) .<_ X )
40 breq1 4656 . . . . . . . . . . 11 |- ( y = ( z ( join ` K ) p ) -> ( y .<_ X <-> ( z ( join ` K ) p ) .<_ X ) )
4140rspcev 3309 . . . . . . . . . 10 |- ( ( ( z ( join ` K ) p ) e. P /\ ( z ( join ` K ) p ) .<_ X ) -> E. y e. P y .<_ X )
4238, 39, 41syl2anc 693 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ ( z e. N /\ z .<_ X /\ p e. A ) /\ ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) ) -> E. y e. P y .<_ X )
43423exp 1264 . . . . . . . 8 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) -> ( ( z e. N /\ z .<_ X /\ p e. A ) -> ( ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) -> E. y e. P y .<_ X ) ) )
44433expd 1284 . . . . . . 7 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) -> ( z e. N -> ( z .<_ X -> ( p e. A -> ( ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) -> E. y e. P y .<_ X ) ) ) ) )
45443imp 1256 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> ( p e. A -> ( ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) -> E. y e. P y .<_ X ) ) )
4645rexlimdv 3030 . . . . 5 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> ( E. p e. A ( z ( <o ` K ) ( z ( join ` K ) p ) /\ ( z ( join ` K ) p ) .<_ X ) -> E. y e. P y .<_ X ) )
4724, 46mpd 15 . . . 4 |- ( ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) /\ z e. N /\ z .<_ X ) -> E. y e. P y .<_ X )
48473exp 1264 . . 3 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) -> ( z e. N -> ( z .<_ X -> E. y e. P y .<_ X ) ) )
4948rexlimdv 3030 . 2 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) -> ( E. z e. N z .<_ X -> E. y e. P y .<_ X ) )
507, 49mpd 15 1 |- ( ( ( K e. HL /\ X e. B ) /\ ( X =/= .0. /\ -. X e. A /\ -. X e. N ) ) -> E. y e. P 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   Latclat 17045   <o ccvr 34549   Atomscatm 34550   HLchlt 34637   LLinesclln 34777   LPlanesclpl 34778
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  df-lplanes 34785
This theorem is referenced by:  lplncvrlvol  34902
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