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Theorem atcvrlln 34316
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 1035 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  K  e.  HL )
2 simpll3 1037 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  Y  e.  B )
3 simpr 461 . . 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 34304 . . 3  |-  ( ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  A )  /\  X C Y )  ->  Y  e.  N
)
101, 2, 3, 4, 9syl31anc 1231 . 2  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  Y  e.  N )
11 simpr 461 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  N )
12 simpll1 1035 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  K  e.  HL )
13 simpll3 1037 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  B )
14 eqid 2467 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
155, 14, 7, 8islln3 34306 . . . . 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 661 . . . 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 1089 . . . . . . 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 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 ) ) )  ->  X  e.  B )
20 simp2l 1022 . . . . . . 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 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 ) ) )  ->  q  e.  A )
22 simp3l 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 ) ) )  ->  p  =/=  q )
23 simp1r 1021 . . . . . . . 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 1025 . . . . . . . 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 4471 . . . . . . 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 34225 . . . . . . 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 1246 . . . . . 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 1195 . . . . 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 2961 . . . 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 465 . . 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 830 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   joincjn 15427    <o ccvr 34059   Atomscatm 34060   HLchlt 34147   LLinesclln 34287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294
This theorem is referenced by:  llncvrlpln  34354  2llnmj  34356  2llnm2N  34364
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