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Theorem atcvrlln 32537
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 1036 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  K  e.  HL )
2 simpll3 1038 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  Y  e.  B )
3 simpr 459 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  X  e.  A )
4 simplr 754 . . 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 32525 . . 3  |-  ( ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  A )  /\  X C Y )  ->  Y  e.  N
)
101, 2, 3, 4, 9syl31anc 1233 . 2  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  Y  e.  N )
11 simpr 459 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  N )
12 simpll1 1036 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  K  e.  HL )
13 simpll3 1038 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  B )
14 eqid 2402 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
155, 14, 7, 8islln3 32527 . . . . 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 659 . . . 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 210 . . 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 1090 . . . . . . 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 1091 . . . . . . 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 1023 . . . . . . 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 1024 . . . . . . 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 1025 . . . . . . 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 1022 . . . . . . . 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 1026 . . . . . . . 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 4419 . . . . . . 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 32446 . . . . . . 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 1248 . . . . . 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 1196 . . . . 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 2902 . . . 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 463 . . 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 833 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  ( X  e.  A  <->  Y  e.  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   joincjn 15897    <o ccvr 32280   Atomscatm 32281   HLchlt 32368   LLinesclln 32508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515
This theorem is referenced by:  llncvrlpln  32575  2llnmj  32577  2llnm2N  32585
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