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Theorem atcvrlln2 30217
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 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 atcvrlln2.n . . . . . 6  |-  N  =  ( LLines `  K )
53, 4llnbase 30207 . . . . 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 2435 . . . . 5  |-  ( join `  K )  =  (
join `  K )
8 atcvrlln2.a . . . . 5  |-  A  =  ( Atoms `  K )
93, 7, 8, 4islln3 30208 . . . 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 4228 . . . . . 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 30131 . . . . . 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 4230 . . . 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 2828 . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459   lecple 13526   joincjn 14391    <o ccvr 29961   Atomscatm 29962   HLchlt 30049   LLinesclln 30189
This theorem is referenced by:  llnexatN  30219  llncmp  30220  2llnmat  30222  2llnmj  30258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-lat 14465  df-clat 14527  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196
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