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

Theorem llnnleat 30310
Description: An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)
Hypotheses
Ref Expression
llnnleat.l  |-  .<_  =  ( le `  K )
llnnleat.a  |-  A  =  ( Atoms `  K )
llnnleat.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llnnleat  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  -.  X  .<_  P )

Proof of Theorem llnnleat
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  X  e.  N )
2 eqid 2436 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2436 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 llnnleat.a . . . . . 6  |-  A  =  ( Atoms `  K )
5 llnnleat.n . . . . . 6  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln 30303 . . . . 5  |-  ( K  e.  HL  ->  ( X  e.  N  <->  ( X  e.  ( Base `  K
)  /\  E. q  e.  A  q (  <o  `  K ) X ) ) )
763ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  ( X  e.  N  <->  ( X  e.  ( Base `  K )  /\  E. q  e.  A  q
(  <o  `  K ) X ) ) )
81, 7mpbid 202 . . 3  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  ( X  e.  (
Base `  K )  /\  E. q  e.  A  q (  <o  `  K
) X ) )
98simprd 450 . 2  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  E. q  e.  A  q (  <o  `  K
) X )
10 simp11 987 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  HL )
11 hlatl 30158 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
1210, 11syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  AtLat )
13 simp2 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q  e.  A )
14 simp13 989 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  P  e.  A )
15 eqid 2436 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
1615, 4atnlt 30111 . . . . 5  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  -.  q ( lt `  K ) P )
1712, 13, 14, 16syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  -.  q ( lt `  K ) P )
182, 4atbase 30087 . . . . . . 7  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
19183ad2ant2 979 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q  e.  ( Base `  K ) )
20 simp12 988 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  X  e.  N )
212, 5llnbase 30306 . . . . . . 7  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
2220, 21syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  X  e.  ( Base `  K ) )
23 simp3 959 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q (  <o  `  K
) X )
242, 15, 3cvrlt 30068 . . . . . 6  |-  ( ( ( K  e.  HL  /\  q  e.  ( Base `  K )  /\  X  e.  ( Base `  K
) )  /\  q
(  <o  `  K ) X )  ->  q
( lt `  K
) X )
2510, 19, 22, 23, 24syl31anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q ( lt `  K ) X )
26 hlpos 30163 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Poset )
2710, 26syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  Poset )
282, 4atbase 30087 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2914, 28syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  P  e.  ( Base `  K ) )
30 llnnleat.l . . . . . . 7  |-  .<_  =  ( le `  K )
312, 30, 15pltletr 14428 . . . . . 6  |-  ( ( K  e.  Poset  /\  (
q  e.  ( Base `  K )  /\  X  e.  ( Base `  K
)  /\  P  e.  ( Base `  K )
) )  ->  (
( q ( lt
`  K ) X  /\  X  .<_  P )  ->  q ( lt
`  K ) P ) )
3227, 19, 22, 29, 31syl13anc 1186 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
( ( q ( lt `  K ) X  /\  X  .<_  P )  ->  q ( lt `  K ) P ) )
3325, 32mpand 657 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
( X  .<_  P  -> 
q ( lt `  K ) P ) )
3417, 33mtod 170 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  -.  X  .<_  P )
3534rexlimdv3a 2832 . 2  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  ( E. q  e.  A  q (  <o  `  K ) X  ->  -.  X  .<_  P ) )
369, 35mpd 15 1  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  -.  X  .<_  P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Posetcpo 14397   ltcplt 14398    <o ccvr 30060   Atomscatm 30061   AtLatcal 30062   HLchlt 30148   LLinesclln 30288
This theorem is referenced by:  llnneat  30311  llnn0  30313  lplnnle2at  30338
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-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-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  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-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-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-poset 14403  df-plt 14415  df-lat 14475  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295
  Copyright terms: Public domain W3C validator