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Theorem lncvrat 33253
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 762 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  ( M `  X )  e.  N
)
2 simpl1 1008 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( ( M `  X )  e.  N  /\  P  .<_  X ) )  ->  K  e.  HL )
3 simpl2 1009 . . . 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 2422 . . . . 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 33247 . . . 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 665 . . 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 213 . 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 1098 . . . . . 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 1100 . . . . . 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 1031 . . . . . 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 1032 . . . . . 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 1033 . . . . . 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 1071 . . . . . . 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 1034 . . . . . . 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 4384 . . . . . 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 32904 . . . . . 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  /\  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 4386 . . . 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 1204 . . 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 2856 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2593   E.wrex 2709   class class class wbr 4359   ` cfv 5537  (class class class)co 6242   Basecbs 15057   lecple 15133   joincjn 16125    <o ccvr 32734   Atomscatm 32735   HLchlt 32822   Linesclines 32965   pmapcpmap 32968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-reu 2715  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-riota 6204  df-ov 6245  df-oprab 6246  df-preset 16109  df-poset 16127  df-plt 16140  df-lub 16156  df-glb 16157  df-join 16158  df-meet 16159  df-p0 16221  df-lat 16228  df-clat 16290  df-oposet 32648  df-ol 32650  df-oml 32651  df-covers 32738  df-ats 32739  df-atl 32770  df-cvlat 32794  df-hlat 32823  df-lines 32972  df-pmap 32975
This theorem is referenced by:  2lnat  33255
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