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Theorem atcvrlln2 33526
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 993 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  X  e.  N
)
2 simpl1 991 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  K  e.  HL )
3 eqid 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 atcvrlln2.n . . . . . 6  |-  N  =  ( LLines `  K )
53, 4llnbase 33516 . . . . 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 2454 . . . . 5  |-  ( join `  K )  =  (
join `  K )
8 atcvrlln2.a . . . . 5  |-  A  =  ( Atoms `  K )
93, 7, 8, 4islln3 33517 . . . 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 1081 . . . . . 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 1082 . . . . . 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 1014 . . . . . 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 1015 . . . . . 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 1016 . . . . . 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 1013 . . . . . . 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 1017 . . . . . . 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 4427 . . . . . 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 33440 . . . . . 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 1237 . . . . 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 4429 . . . 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 1187 . . 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 2953 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237    <o ccvr 33270   Atomscatm 33271   HLchlt 33358   LLinesclln 33498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-llines 33505
This theorem is referenced by:  llnexatN  33528  llncmp  33529  2llnmat  33531  2llnmj  33567
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