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Theorem lplnle 29729
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 29707 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A ) )  ->  E. z  e.  N  z  .<_  X )
763adantr3 1116 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/= 
.0.  /\  -.  X  e.  A  /\  -.  X  e.  N ) )  ->  E. z  e.  N  z  .<_  X )
8 simp1ll 1018 . . . . . 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 29698 . . . . . . 7  |-  ( z  e.  N  ->  z  e.  B )
1093ad2ant2 977 . . . . . 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 1019 . . . . . 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 957 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N
) )  /\  z  e.  N  /\  z  .<_  X )  ->  z  .<_  X )
13 simp2 956 . . . . . . . 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 1053 . . . . . . . 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 2536 . . . . . . . 8  |-  ( ( z  e.  N  /\  -.  X  e.  N
)  ->  z  =/=  X )
1613, 14, 15syl2anc 642 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( X  =/=  .0.  /\  -.  X  e.  A  /\  -.  X  e.  N
) )  /\  z  e.  N  /\  z  .<_  X )  ->  z  =/=  X )
17 eqid 2283 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
182, 17pltval 14094 . . . . . . . 8  |-  ( ( K  e.  HL  /\  z  e.  N  /\  X  e.  B )  ->  ( z ( lt
`  K ) X  <-> 
( z  .<_  X  /\  z  =/=  X ) ) )
198, 13, 11, 18syl3anc 1182 . . . . . . 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 888 . . . . . 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 2283 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
22 eqid 2283 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
231, 2, 17, 21, 22, 4hlrelat3 29601 . . . . . 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 1185 . . . . 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 1018 . . . . . . . . . . 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 29553 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  Lat )
2725, 26syl 15 . . . . . . . . . . . 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 988 . . . . . . . . . . . . 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 15 . . . . . . . . . . . 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 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 ) )  ->  p  e.  A )
311, 4atbase 29479 . . . . . . . . . . . . 13  |-  ( p  e.  A  ->  p  e.  B )
3230, 31syl 15 . . . . . . . . . . . 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 14156 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  z  e.  B  /\  p  e.  B )  ->  ( z ( join `  K ) p )  e.  B )
3427, 29, 32, 33syl3anc 1182 . . . . . . . . . . 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 983 . . . . . . . . . . 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 29721 . . . . . . . . . . 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 1185 . . . . . . . . . 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 984 . . . . . . . . . 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 4026 . . . . . . . . . . 11  |-  ( y  =  ( z (
join `  K )
p )  ->  (
y  .<_  X  <->  ( z
( join `  K )
p )  .<_  X ) )
4140rspcev 2884 . . . . . . . . . 10  |-  ( ( ( z ( join `  K ) p )  e.  P  /\  (
z ( join `  K
) p )  .<_  X )  ->  E. y  e.  P  y  .<_  X )
4238, 39, 41syl2anc 642 . . . . . . . . 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 1150 . . . . . . . 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 1168 . . . . . . 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 1145 . . . . . 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 2666 . . . . 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 14 . . . 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 1150 . . 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 2666 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   ltcplt 14075   joincjn 14078   0.cp0 14143   Latclat 14151    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  lplncvrlvol  29805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688
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