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

Theorem llnnleat 34526
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 997 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  X  e.  N )
2 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2467 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 llnnleat.a . . . . . 6  |-  A  =  ( Atoms `  K )
5 llnnleat.n . . . . . 6  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln 34519 . . . . 5  |-  ( K  e.  HL  ->  ( X  e.  N  <->  ( X  e.  ( Base `  K
)  /\  E. q  e.  A  q (  <o  `  K ) X ) ) )
763ad2ant1 1017 . . . 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 210 . . 3  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  ( X  e.  (
Base `  K )  /\  E. q  e.  A  q (  <o  `  K
) X ) )
98simprd 463 . 2  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  E. q  e.  A  q (  <o  `  K
) X )
10 simp11 1026 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  HL )
11 hlatl 34374 . . . . . 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 997 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q  e.  A )
14 simp13 1028 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  P  e.  A )
15 eqid 2467 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
1615, 4atnlt 34327 . . . . 5  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  -.  q ( lt `  K ) P )
1712, 13, 14, 16syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  -.  q ( lt `  K ) P )
182, 4atbase 34303 . . . . . . 7  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
19183ad2ant2 1018 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q  e.  ( Base `  K ) )
20 simp12 1027 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  X  e.  N )
212, 5llnbase 34522 . . . . . . 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 998 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q (  <o  `  K
) X )
242, 15, 3cvrlt 34284 . . . . . 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 1231 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q ( lt `  K ) X )
26 hlpos 34379 . . . . . . 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 34303 . . . . . . 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 15461 . . . . . 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 1230 . . . . 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 675 . . . 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 177 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  -.  X  .<_  P )
3534rexlimdv3a 2957 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447   ` cfv 5588   Basecbs 14493   lecple 14565   Posetcpo 15430   ltcplt 15431    <o ccvr 34276   Atomscatm 34277   AtLatcal 34278   HLchlt 34364   LLinesclln 34504
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-poset 15436  df-plt 15448  df-glb 15465  df-p0 15529  df-lat 15536  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511
This theorem is referenced by:  llnneat  34527  llnn0  34529  lplnnle2at  34554
  Copyright terms: Public domain W3C validator