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Theorem llnnleat 29702
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 956 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  X  e.  N )
2 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2283 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 llnnleat.a . . . . . 6  |-  A  =  ( Atoms `  K )
5 llnnleat.n . . . . . 6  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln 29695 . . . . 5  |-  ( K  e.  HL  ->  ( X  e.  N  <->  ( X  e.  ( Base `  K
)  /\  E. q  e.  A  q (  <o  `  K ) X ) ) )
763ad2ant1 976 . . . 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 201 . . 3  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  ( X  e.  (
Base `  K )  /\  E. q  e.  A  q (  <o  `  K
) X ) )
98simprd 449 . 2  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  E. q  e.  A  q (  <o  `  K
) X )
10 simp11 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  HL )
11 hlatl 29550 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
1210, 11syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  AtLat )
13 simp2 956 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q  e.  A )
14 simp13 987 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  P  e.  A )
15 eqid 2283 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
1615, 4atnlt 29503 . . . . 5  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  -.  q ( lt `  K ) P )
1712, 13, 14, 16syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  -.  q ( lt `  K ) P )
182, 4atbase 29479 . . . . . . 7  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
19183ad2ant2 977 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q  e.  ( Base `  K ) )
20 simp12 986 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  X  e.  N )
212, 5llnbase 29698 . . . . . . 7  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
2220, 21syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  X  e.  ( Base `  K ) )
23 simp3 957 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q (  <o  `  K
) X )
242, 15, 3cvrlt 29460 . . . . . 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 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q ( lt `  K ) X )
26 hlpos 29555 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Poset )
2710, 26syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  Poset )
282, 4atbase 29479 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2914, 28syl 15 . . . . . 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 14105 . . . . . 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 1184 . . . . 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 656 . . . 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 168 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  -.  X  .<_  P )
3534rexlimdv3a 2669 . 2  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  ( E. q  e.  A  q (  <o  `  K ) X  ->  -.  X  .<_  P ) )
369, 35mpd 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LLinesclln 29680
This theorem is referenced by:  llnneat  29703  llnn0  29705  lplnnle2at  29730
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-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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687
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