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Theorem atcvrlln 29709
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 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  K  e.  HL )
2 simpll3 996 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  Y  e.  B )
3 simpr 447 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  X  e.  A )
4 simplr 731 . . 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 29697 . . 3  |-  ( ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  A )  /\  X C Y )  ->  Y  e.  N
)
101, 2, 3, 4, 9syl31anc 1185 . 2  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  Y  e.  N )
11 simpr 447 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  N )
12 simpll1 994 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  K  e.  HL )
13 simpll3 996 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  B )
14 eqid 2283 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
155, 14, 7, 8islln3 29699 . . . . 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 642 . . . 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 201 . . 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 1048 . . . . . . 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 1049 . . . . . . 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 981 . . . . . . 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 982 . . . . . . 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 983 . . . . . . 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 980 . . . . . . . 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 984 . . . . . . . 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 4047 . . . . . . 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 29618 . . . . . . 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 1200 . . . . . 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 1150 . . . . 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 2673 . . . 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 451 . . 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 14 . 2  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  X  e.  A )
3210, 31impbida 805 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   LLinesclln 29680
This theorem is referenced by:  llncvrlpln  29747  2llnmj  29749  2llnm2N  29757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687
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