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Theorem lplnle 30351
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 30329 . . 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 30320 . . . . . . 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 2549 . . . . . . . 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 2296 . . . . . . . . 9  |-  ( lt
`  K )  =  ( lt `  K
)
182, 17pltval 14110 . . . . . . . 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 2296 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
22 eqid 2296 . . . . . . 7  |-  (  <o  `  K )  =  ( 
<o  `  K )
231, 2, 17, 21, 22, 4hlrelat3 30223 . . . . . 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 30175 . . . . . . . . . . . . 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 30101 . . . . . . . . . . . . 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 14172 . . . . . . . . . . . 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 30343 . . . . . . . . . . 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 4042 . . . . . . . . . . 11  |-  ( y  =  ( z (
join `  K )
p )  ->  (
y  .<_  X  <->  ( z
( join `  K )
p )  .<_  X ) )
4140rspcev 2897 . . . . . . . . . 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 2679 . . . . 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 2679 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   ltcplt 14091   joincjn 14094   0.cp0 14159   Latclat 14167    <o ccvr 30074   Atomscatm 30075   HLchlt 30162   LLinesclln 30302   LPlanesclpl 30303
This theorem is referenced by:  lplncvrlvol  30427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310
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