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Theorem llnnleat 32831
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 1006 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  X  e.  N )
2 eqid 2420 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2420 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 llnnleat.a . . . . . 6  |-  A  =  ( Atoms `  K )
5 llnnleat.n . . . . . 6  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln 32824 . . . . 5  |-  ( K  e.  HL  ->  ( X  e.  N  <->  ( X  e.  ( Base `  K
)  /\  E. q  e.  A  q (  <o  `  K ) X ) ) )
763ad2ant1 1026 . . . 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 213 . . 3  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  ( X  e.  (
Base `  K )  /\  E. q  e.  A  q (  <o  `  K
) X ) )
98simprd 464 . 2  |-  ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  ->  E. q  e.  A  q (  <o  `  K
) X )
10 simp11 1035 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  HL )
11 hlatl 32679 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
1210, 11syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  AtLat )
13 simp2 1006 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q  e.  A )
14 simp13 1037 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  P  e.  A )
15 eqid 2420 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
1615, 4atnlt 32632 . . . . 5  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  -.  q ( lt `  K ) P )
1712, 13, 14, 16syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  -.  q ( lt `  K ) P )
182, 4atbase 32608 . . . . . . 7  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
19183ad2ant2 1027 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q  e.  ( Base `  K ) )
20 simp12 1036 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  X  e.  N )
212, 5llnbase 32827 . . . . . . 7  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
2220, 21syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  X  e.  ( Base `  K ) )
23 simp3 1007 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q (  <o  `  K
) X )
242, 15, 3cvrlt 32589 . . . . . 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 1267 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  -> 
q ( lt `  K ) X )
26 hlpos 32684 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Poset )
2710, 26syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  K  e.  Poset )
282, 4atbase 32608 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2914, 28syl 17 . . . . . 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 16169 . . . . . 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 1266 . . . . 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 679 . . . 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 180 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  P  e.  A )  /\  q  e.  A  /\  q (  <o  `  K
) X )  ->  -.  X  .<_  P )
3534rexlimdv3a 2917 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   E.wrex 2774   class class class wbr 4417   ` cfv 5592   Basecbs 15081   lecple 15157   Posetcpo 16137   ltcplt 16138    <o ccvr 32581   Atomscatm 32582   AtLatcal 32583   HLchlt 32669   LLinesclln 32809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-preset 16125  df-poset 16143  df-plt 16156  df-glb 16173  df-p0 16237  df-lat 16244  df-covers 32585  df-ats 32586  df-atl 32617  df-cvlat 32641  df-hlat 32670  df-llines 32816
This theorem is referenced by:  llnneat  32832  llnn0  32834  lplnnle2at  32859
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