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Theorem lplnle 34629
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 34607 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. z  e.  N  z  .<_  X )
763adantr3 1157 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A  /\  -.  X  e.  N ) )  ->  E. z  e.  N  z  .<_  X )
8 simp1ll 1059 . . . . . 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 34598 . . . . . . 7  |-  ( z  e.  N  ->  z  e.  B )
1093ad2ant2 1018 . . . . . 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 1060 . . . . . 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 998 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N
) )  /\  z  e.  N  /\  z  .<_  X )  ->  z  .<_  X )
13 simp2 997 . . . . . . . 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 1094 . . . . . . . 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 2797 . . . . . . . 8  |-  ( ( z  e.  N  /\  -.  X  e.  N
)  ->  z  =/=  X )
1613, 14, 15syl2anc 661 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N
) )  /\  z  e.  N  /\  z  .<_  X )  ->  z  =/=  X )
17 eqid 2467 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
182, 17pltval 15459 . . . . . . . 8  |-  ( ( K  e.  HL  /\  z  e.  N  /\  X  e.  B )  ->  ( z ( lt
`  K ) X  <-> 
( z  .<_  X  /\  z  =/=  X ) ) )
198, 13, 11, 18syl3anc 1228 . . . . . . 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 920 . . . . . 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 2467 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
22 eqid 2467 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
231, 2, 17, 21, 22, 4hlrelat3 34501 . . . . . 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 1231 . . . . 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 1059 . . . . . . . . . . 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 34453 . . . . . . . . . . . . 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 1029 . . . . . . . . . . . . 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 1031 . . . . . . . . . . . . 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 34379 . . . . . . . . . . . . 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 15550 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  p  e.  B )  ->  ( z ( join `  K ) p )  e.  B )
3427, 29, 32, 33syl3anc 1228 . . . . . . . . . . 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 1024 . . . . . . . . . . 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 34621 . . . . . . . . . . 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 1231 . . . . . . . . . 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 1025 . . . . . . . . . 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 4455 . . . . . . . . . . 11  |-  ( y  =  ( z (
join `  K )
p )  ->  (
y  .<_  X  <->  ( z
( join `  K )
p )  .<_  X ) )
4140rspcev 3219 . . . . . . . . . 10  |-  ( ( ( z ( join `  K ) p )  e.  P  /\  (
z ( join `  K
) p )  .<_  X )  ->  E. y  e.  P  y  .<_  X )
4238, 39, 41syl2anc 661 . . . . . . . . 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 1195 . . . . . . . 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 1213 . . . . . . 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 1190 . . . . . 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 2957 . . . . 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 1195 . . 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 2957 . 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   Basecbs 14502   lecple 14574   ltcplt 15440   joincjn 15443   0.cp0 15536   Latclat 15544    <o ccvr 34352   Atomscatm 34353   HLchlt 34440   LLinesclln 34580   LPlanesclpl 34581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-poset 15445  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-clat 15607  df-oposet 34266  df-ol 34268  df-oml 34269  df-covers 34356  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441  df-llines 34587  df-lplanes 34588
This theorem is referenced by:  lplncvrlvol  34705
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