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Theorem atcvrlln2 34716
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 1001 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  X  e.  N
)
2 simpl1 999 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  K  e.  HL )
3 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 atcvrlln2.n . . . . . 6  |-  N  =  ( LLines `  K )
53, 4llnbase 34706 . . . . 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 2467 . . . . 5  |-  ( join `  K )  =  (
join `  K )
8 atcvrlln2.a . . . . 5  |-  A  =  ( Atoms `  K )
93, 7, 8, 4islln3 34707 . . . 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 661 . . 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 210 . 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 1089 . . . . . 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 1090 . . . . . 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 1022 . . . . . 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 1023 . . . . . 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 1024 . . . . . 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 1021 . . . . . . 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 1025 . . . . . . 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 4477 . . . . . 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 34630 . . . . . 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 1246 . . . . 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 4479 . . . 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 1195 . . 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 2965 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448    <o ccvr 34460   Atomscatm 34461   HLchlt 34548   LLinesclln 34688
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695
This theorem is referenced by:  llnexatN  34718  llncmp  34719  2llnmat  34721  2llnmj  34757
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