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Theorem lplnle 32814
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 32792 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. z  e.  N  z  .<_  X )
763adantr3 1166 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A  /\  -.  X  e.  N ) )  ->  E. z  e.  N  z  .<_  X )
8 simp1ll 1068 . . . . . 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 32783 . . . . . . 7  |-  ( z  e.  N  ->  z  e.  B )
1093ad2ant2 1027 . . . . . 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 1069 . . . . . 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 1007 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N
) )  /\  z  e.  N  /\  z  .<_  X )  ->  z  .<_  X )
13 simp2 1006 . . . . . . . 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 1103 . . . . . . . 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 2761 . . . . . . . 8  |-  ( ( z  e.  N  /\  -.  X  e.  N
)  ->  z  =/=  X )
1613, 14, 15syl2anc 665 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N
) )  /\  z  e.  N  /\  z  .<_  X )  ->  z  =/=  X )
17 eqid 2429 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
182, 17pltval 16157 . . . . . . . 8  |-  ( ( K  e.  HL  /\  z  e.  N  /\  X  e.  B )  ->  ( z ( lt
`  K ) X  <-> 
( z  .<_  X  /\  z  =/=  X ) ) )
198, 13, 11, 18syl3anc 1264 . . . . . . 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 930 . . . . . 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 2429 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
22 eqid 2429 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
231, 2, 17, 21, 22, 4hlrelat3 32686 . . . . . 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 1267 . . . . 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 1068 . . . . . . . . . . 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 32638 . . . . . . . . . . . . 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 1038 . . . . . . . . . . . . 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 1040 . . . . . . . . . . . . 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 32564 . . . . . . . . . . . . 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 16248 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  p  e.  B )  ->  ( z ( join `  K ) p )  e.  B )
3427, 29, 32, 33syl3anc 1264 . . . . . . . . . . 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 1033 . . . . . . . . . . 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 32806 . . . . . . . . . . 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 1267 . . . . . . . . . 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 1034 . . . . . . . . . 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 4429 . . . . . . . . . . 11  |-  ( y  =  ( z (
join `  K )
p )  ->  (
y  .<_  X  <->  ( z
( join `  K )
p )  .<_  X ) )
4140rspcev 3188 . . . . . . . . . 10  |-  ( ( ( z ( join `  K ) p )  e.  P  /\  (
z ( join `  K
) p )  .<_  X )  ->  E. y  e.  P  y  .<_  X )
4238, 39, 41syl2anc 665 . . . . . . . . 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 1204 . . . . . . . 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 1222 . . . . . . 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 1199 . . . . . 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 2922 . . . . 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 1204 . . 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 2922 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   ltcplt 16137   joincjn 16140   0.cp0 16234   Latclat 16242    <o ccvr 32537   Atomscatm 32538   HLchlt 32625   LLinesclln 32765   LPlanesclpl 32766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773
This theorem is referenced by:  lplncvrlvol  32890
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