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

Theorem lncvrat 32763
Description: A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)
Hypotheses
Ref Expression
lncvrat.b  |-  B  =  ( Base `  K
)
lncvrat.l  |-  .<_  =  ( le `  K )
lncvrat.c  |-  C  =  (  <o  `  K )
lncvrat.a  |-  A  =  ( Atoms `  K )
lncvrat.n  |-  N  =  ( Lines `  K )
lncvrat.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
lncvrat  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  P C X )

Proof of Theorem lncvrat
Dummy variables  r 
q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 755 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  ( M `  X )  e.  N
)
2 simpl1 998 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  K  e.  HL )
3 simpl2 999 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  X  e.  B )
4 lncvrat.b . . . . 5  |-  B  =  ( Base `  K
)
5 eqid 2400 . . . . 5  |-  ( join `  K )  =  (
join `  K )
6 lncvrat.a . . . . 5  |-  A  =  ( Atoms `  K )
7 lncvrat.n . . . . 5  |-  N  =  ( Lines `  K )
8 lncvrat.m . . . . 5  |-  M  =  ( pmap `  K
)
94, 5, 6, 7, 8isline3 32757 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) ) )
102, 3, 9syl2anc 659 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  ( ( M `  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  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  ( q ( join `  K ) r ) ) )
12 simp1l1 1088 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  ( q  e.  A  /\  r  e.  A )  /\  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) )  ->  K  e.  HL )
13 simp1l3 1090 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  ( q  e.  A  /\  r  e.  A )  /\  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) )  ->  P  e.  A
)
14 simp2l 1021 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  ( q  e.  A  /\  r  e.  A )  /\  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) )  ->  q  e.  A
)
15 simp2r 1022 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  ( q  e.  A  /\  r  e.  A )  /\  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) )  ->  r  e.  A
)
16 simp3l 1023 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  ( q  e.  A  /\  r  e.  A )  /\  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) )  ->  q  =/=  r
)
17 simp1rr 1061 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  ( q  e.  A  /\  r  e.  A )  /\  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) )  ->  P  .<_  X )
18 simp3r 1024 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  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 4416 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  ( q  e.  A  /\  r  e.  A )  /\  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) )  ->  P  .<_  ( q ( join `  K
) r ) )
20 lncvrat.l . . . . . . 7  |-  .<_  =  ( le `  K )
21 lncvrat.c . . . . . . 7  |-  C  =  (  <o  `  K )
2220, 5, 21, 6atcvrj2 32414 . . . . . 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  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  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 4418 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  (
( M `  X
)  e.  N  /\  P  .<_  X ) )  /\  ( q  e.  A  /\  r  e.  A )  /\  (
q  =/=  r  /\  X  =  ( q
( join `  K )
r ) ) )  ->  P C X )
25243exp 1194 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  ( (
q  e.  A  /\  r  e.  A )  ->  ( ( q  =/=  r  /\  X  =  ( q ( join `  K ) r ) )  ->  P C X ) ) )
2625rexlimdvv 2899 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  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  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  P C X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   E.wrex 2752   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787    <o ccvr 32244   Atomscatm 32245   HLchlt 32332   Linesclines 32475   pmapcpmap 32478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-lines 32482  df-pmap 32485
This theorem is referenced by:  2lnat  32765
  Copyright terms: Public domain W3C validator