Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lplnle Structured version   Unicode version

Theorem lplnle 30337
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 30315 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. z  e.  N  z  .<_  X )
763adantr3 1118 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A  /\  -.  X  e.  N ) )  ->  E. z  e.  N  z  .<_  X )
8 simp1ll 1020 . . . . . 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 30306 . . . . . . 7  |-  ( z  e.  N  ->  z  e.  B )
1093ad2ant2 979 . . . . . 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 1021 . . . . . 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 959 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N
) )  /\  z  e.  N  /\  z  .<_  X )  ->  z  .<_  X )
13 simp2 958 . . . . . . . 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 1055 . . . . . . . 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 2694 . . . . . . . 8  |-  ( ( z  e.  N  /\  -.  X  e.  N
)  ->  z  =/=  X )
1613, 14, 15syl2anc 643 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N
) )  /\  z  e.  N  /\  z  .<_  X )  ->  z  =/=  X )
17 eqid 2436 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
182, 17pltval 14417 . . . . . . . 8  |-  ( ( K  e.  HL  /\  z  e.  N  /\  X  e.  B )  ->  ( z ( lt
`  K ) X  <-> 
( z  .<_  X  /\  z  =/=  X ) ) )
198, 13, 11, 18syl3anc 1184 . . . . . . 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 889 . . . . . 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 2436 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
22 eqid 2436 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
231, 2, 17, 21, 22, 4hlrelat3 30209 . . . . . 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 1187 . . . . 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 1020 . . . . . . . . . . 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 30161 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  Lat )
2725, 26syl 16 . . . . . . . . . . . 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 990 . . . . . . . . . . . . 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 16 . . . . . . . . . . . 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 992 . . . . . . . . . . . . 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 30087 . . . . . . . . . . . . 13  |-  ( p  e.  A  ->  p  e.  B )
3230, 31syl 16 . . . . . . . . . . . 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 14479 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  p  e.  B )  ->  ( z ( join `  K ) p )  e.  B )
3427, 29, 32, 33syl3anc 1184 . . . . . . . . . . 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 985 . . . . . . . . . . 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 30329 . . . . . . . . . . 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 1187 . . . . . . . . . 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 986 . . . . . . . . . 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 4215 . . . . . . . . . . 11  |-  ( y  =  ( z (
join `  K )
p )  ->  (
y  .<_  X  <->  ( z
( join `  K )
p )  .<_  X ) )
4140rspcev 3052 . . . . . . . . . 10  |-  ( ( ( z ( join `  K ) p )  e.  P  /\  (
z ( join `  K
) p )  .<_  X )  ->  E. y  e.  P  y  .<_  X )
4238, 39, 41syl2anc 643 . . . . . . . . 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 1152 . . . . . . . 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 1170 . . . . . . 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 1147 . . . . . 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 2829 . . . . 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 1152 . . 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 2829 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   ltcplt 14398   joincjn 14401   0.cp0 14466   Latclat 14474    <o ccvr 30060   Atomscatm 30061   HLchlt 30148   LLinesclln 30288   LPlanesclpl 30289
This theorem is referenced by:  lplncvrlvol  30413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296
  Copyright terms: Public domain W3C validator