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Theorem atcvrlln2 29635
Description: An atom under a line is covered by it. (Contributed by NM, 2-Jul-2012.)
Hypotheses
Ref Expression
atcvrlln2.l  |-  .<_  =  ( le `  K )
atcvrlln2.c  |-  C  =  (  <o  `  K )
atcvrlln2.a  |-  A  =  ( Atoms `  K )
atcvrlln2.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
atcvrlln2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  P C X )

Proof of Theorem atcvrlln2
Dummy variables  r 
q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 962 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  X  e.  N
)
2 simpl1 960 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  K  e.  HL )
3 eqid 2389 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 atcvrlln2.n . . . . . 6  |-  N  =  ( LLines `  K )
53, 4llnbase 29625 . . . . 5  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
61, 5syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  X  e.  (
Base `  K )
)
7 eqid 2389 . . . . 5  |-  ( join `  K )  =  (
join `  K )
8 atcvrlln2.a . . . . 5  |-  A  =  ( Atoms `  K )
93, 7, 8, 4islln3 29626 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K ) )  -> 
( X  e.  N  <->  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) ) )
102, 6, 9syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  ( X  e.  N  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) ) )
111, 10mpbid 202 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )
12 simp1l1 1050 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  K  e.  HL )
13 simp1l2 1051 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  P  e.  A )
14 simp2l 983 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  q  e.  A )
15 simp2r 984 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  r  e.  A )
16 simp3l 985 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  q  =/=  r )
17 simp1r 982 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  P  .<_  X )
18 simp3r 986 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  X  =  ( q ( join `  K ) r ) )
1917, 18breqtrd 4179 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  P  .<_  ( q ( join `  K
) r ) )
20 atcvrlln2.l . . . . . . 7  |-  .<_  =  ( le `  K )
21 atcvrlln2.c . . . . . . 7  |-  C  =  (  <o  `  K )
2220, 7, 21, 8atcvrj2 29549 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  q  e.  A  /\  r  e.  A
)  /\  ( q  =/=  r  /\  P  .<_  ( q ( join `  K
) r ) ) )  ->  P C
( q ( join `  K ) r ) )
2312, 13, 14, 15, 16, 19, 22syl132anc 1202 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  P C
( q ( join `  K ) r ) )
2423, 18breqtrrd 4181 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  /\  (
q  e.  A  /\  r  e.  A )  /\  ( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) ) )  ->  P C X )
25243exp 1152 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  ( ( q  e.  A  /\  r  e.  A )  ->  (
( q  =/=  r  /\  X  =  (
q ( join `  K
) r ) )  ->  P C X ) ) )
2625rexlimdvv 2781 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  ( E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  ( q ( join `  K ) r ) )  ->  P C X ) )
2711, 26mpd 15 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  P C X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330    <o ccvr 29379   Atomscatm 29380   HLchlt 29467   LLinesclln 29607
This theorem is referenced by:  llnexatN  29637  llncmp  29638  2llnmat  29640  2llnmj  29676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614
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