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Theorem atcvrlln 29780
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 995 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  X  e.  A )  ->  K  e.  HL )
2 simpll3 997 . . 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 29768 . . 3  |-  ( ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  A )  /\  X C Y )  ->  Y  e.  N
)
101, 2, 3, 4, 9syl31anc 1186 . 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 995 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  K  e.  HL )
13 simpll3 997 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  /\  Y  e.  N )  ->  Y  e.  B )
14 eqid 2366 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
155, 14, 7, 8islln3 29770 . . . . 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 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 ) ) )  ->  K  e.  HL )
19 simp1l2 1050 . . . . . . 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 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 ) ) )  ->  p  e.  A )
21 simp2r 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 ) ) )  ->  q  e.  A )
22 simp3l 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  =/=  q )
23 simp1r 981 . . . . . . . 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 985 . . . . . . . 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 4149 . . . . . . 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 29689 . . . . . . 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 1201 . . . . . 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 1151 . . . . 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 2758 . . . 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   joincjn 14288    <o ccvr 29523   Atomscatm 29524   HLchlt 29611   LLinesclln 29751
This theorem is referenced by:  llncvrlpln  29818  2llnmj  29820  2llnm2N  29828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758
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