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Theorem llnnleat 30324
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 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2296 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
4 llnnleat.a . . . . . 6  |-  A  =  ( Atoms `  K )
5 llnnleat.n . . . . . 6  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln 30317 . . . . 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 30172 . . . . . 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 2296 . . . . . 6  |-  ( lt
`  K )  =  ( lt `  K
)
1615, 4atnlt 30125 . . . . 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 30101 . . . . . . 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 30320 . . . . . . 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 30082 . . . . . 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 30177 . . . . . . 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 30101 . . . . . . 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 14121 . . . . . 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 2682 . 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 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Posetcpo 14090   ltcplt 14091    <o ccvr 30074   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   LLinesclln 30302
This theorem is referenced by:  llnneat  30325  llnn0  30327  lplnnle2at  30352
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-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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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-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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-poset 14096  df-plt 14108  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309
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