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Theorem atcvrlln 30415
Description: An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.)
Hypotheses
Ref Expression
atcvrlln.b  |-  B  =  ( Base `  K
)
atcvrlln.c  |-  C  =  (  <o  `  K )
atcvrlln.a  |-  A  =  ( Atoms `  K )
atcvrlln.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
atcvrlln  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( X  e.  A  <->  Y  e.  N
) )

Proof of Theorem atcvrlln
Dummy variables  q  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll1 997 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  K  e.  HL )
2 simpll3 999 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  Y  e.  B )
3 simpr 449 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  X  e.  A )
4 simplr 733 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  X C Y )
5 atcvrlln.b . . . 4  |-  B  =  ( Base `  K
)
6 atcvrlln.c . . . 4  |-  C  =  (  <o  `  K )
7 atcvrlln.a . . . 4  |-  A  =  ( Atoms `  K )
8 atcvrlln.n . . . 4  |-  N  =  ( LLines `  K )
95, 6, 7, 8llni 30403 . . 3  |-  ( ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  A )  /\  X C Y )  ->  Y  e.  N
)
101, 2, 3, 4, 9syl31anc 1188 . 2  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  Y  e.  N )
11 simpr 449 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  N )
12 simpll1 997 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  K  e.  HL )
13 simpll3 999 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  B )
14 eqid 2442 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
155, 14, 7, 8islln3 30405 . . . . 5  |-  ( ( K  e.  HL  /\  Y  e.  B )  ->  ( Y  e.  N  <->  E. p  e.  A  E. q  e.  A  (
p  =/=  q  /\  Y  =  ( p
( join `  K )
q ) ) ) )
1612, 13, 15syl2anc 644 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  ( Y  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  Y  =  ( p ( join `  K ) q ) ) ) )
1711, 16mpbid 203 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  Y  =  ( p ( join `  K ) q ) ) )
18 simp1l1 1051 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  K  e.  HL )
19 simp1l2 1052 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  X  e.  B )
20 simp2l 984 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  p  e.  A )
21 simp2r 985 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  q  e.  A )
22 simp3l 986 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  p  =/=  q )
23 simp1r 983 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  X C Y )
24 simp3r 987 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  Y  =  ( p ( join `  K ) q ) )
2523, 24breqtrd 4261 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  X C
( p ( join `  K ) q ) )
265, 14, 6, 7cvrat2 30324 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  p  e.  A  /\  q  e.  A
)  /\  ( p  =/=  q  /\  X C ( p ( join `  K ) q ) ) )  ->  X  e.  A )
2718, 19, 20, 21, 22, 25, 26syl132anc 1203 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  (
p  e.  A  /\  q  e.  A )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  X  e.  A )
28273exp 1153 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( ( p  e.  A  /\  q  e.  A )  ->  (
( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) )  ->  X  e.  A
) ) )
2928rexlimdvv 2842 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  Y  =  ( p ( join `  K ) q ) )  ->  X  e.  A ) )
3029adantr 453 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  ( E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) )  ->  X  e.  A
) )
3117, 30mpd 15 . 2  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  X  e.  A )
3210, 31impbida 807 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( X  e.  A  <->  Y  e.  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   E.wrex 2712   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   joincjn 14432    <o ccvr 30158   Atomscatm 30159   HLchlt 30246   LLinesclln 30386
This theorem is referenced by:  llncvrlpln  30453  2llnmj  30455  2llnm2N  30463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-llines 30393
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