Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atcvrlln2 Structured version   Unicode version

Theorem atcvrlln2 33002
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 1010 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  X  e.  N
)
2 simpl1 1008 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  K  e.  HL )
3 eqid 2422 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4 atcvrlln2.n . . . . . 6  |-  N  =  ( LLines `  K )
53, 4llnbase 32992 . . . . 5  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
61, 5syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  N )  /\  P  .<_  X )  ->  X  e.  (
Base `  K )
)
7 eqid 2422 . . . . 5  |-  ( join `  K )  =  (
join `  K )
8 atcvrlln2.a . . . . 5  |-  A  =  ( Atoms `  K )
93, 7, 8, 4islln3 32993 . . . 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 665 . . 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 213 . 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 1098 . . . . . 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 1099 . . . . . 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 1031 . . . . . 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 1032 . . . . . 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 1033 . . . . . 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 1030 . . . . . . 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 1034 . . . . . . 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 4445 . . . . . 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 32916 . . . . . 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 1282 . . . . 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 4447 . . . 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 1204 . . 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 2923 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776   class class class wbr 4420   ` cfv 5597  (class class class)co 6301   Basecbs 15108   lecple 15184   joincjn 16176    <o ccvr 32746   Atomscatm 32747   HLchlt 32834   LLinesclln 32974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-preset 16160  df-poset 16178  df-plt 16191  df-lub 16207  df-glb 16208  df-join 16209  df-meet 16210  df-p0 16272  df-lat 16279  df-clat 16341  df-oposet 32660  df-ol 32662  df-oml 32663  df-covers 32750  df-ats 32751  df-atl 32782  df-cvlat 32806  df-hlat 32835  df-llines 32981
This theorem is referenced by:  llnexatN  33004  llncmp  33005  2llnmat  33007  2llnmj  33043
  Copyright terms: Public domain W3C validator