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Theorem lncvrelatN 35957
Description: A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lncvrelat.b  |-  B  =  ( Base `  K
)
lncvrelat.c  |-  C  =  (  <o  `  K )
lncvrelat.a  |-  A  =  ( Atoms `  K )
lncvrelat.n  |-  N  =  ( Lines `  K )
lncvrelat.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
lncvrelatN  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  ( ( M `  X )  e.  N  /\  P C X ) )  ->  P  e.  A )

Proof of Theorem lncvrelatN
Dummy variables  r 
q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 35540 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1015 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  ->  K  e.  Lat )
3 eqid 2396 . . . . 5  |-  ( join `  K )  =  (
join `  K )
4 lncvrelat.a . . . . 5  |-  A  =  ( Atoms `  K )
5 lncvrelat.n . . . . 5  |-  N  =  ( Lines `  K )
6 lncvrelat.m . . . . 5  |-  M  =  ( pmap `  K
)
73, 4, 5, 6isline2 35950 . . . 4  |-  ( K  e.  Lat  ->  (
( M `  X
)  e.  N  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  ( M `  X )  =  ( M `  ( q ( join `  K ) r ) ) ) ) )
82, 7syl 16 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  ( M `  X )  =  ( M `  ( q ( join `  K ) r ) ) ) ) )
9 simpll1 1033 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  K  e.  HL )
10 simpll2 1034 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  X  e.  B )
119, 1syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  K  e.  Lat )
12 simplrl 759 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  q  e.  A )
13 lncvrelat.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
1413, 4atbase 35466 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
1512, 14syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  q  e.  B )
16 simplrr 760 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  r  e.  A )
1713, 4atbase 35466 . . . . . . . . 9  |-  ( r  e.  A  ->  r  e.  B )
1816, 17syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  r  e.  B )
1913, 3latjcl 15821 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  q  e.  B  /\  r  e.  B )  ->  ( q ( join `  K ) r )  e.  B )
2011, 15, 18, 19syl3anc 1226 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  (
q ( join `  K
) r )  e.  B )
2113, 6pmap11 35938 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( q ( join `  K ) r )  e.  B )  -> 
( ( M `  X )  =  ( M `  ( q ( join `  K
) r ) )  <-> 
X  =  ( q ( join `  K
) r ) ) )
229, 10, 20, 21syl3anc 1226 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  (
( M `  X
)  =  ( M `
 ( q (
join `  K )
r ) )  <->  X  =  ( q ( join `  K ) r ) ) )
23 breq2 4388 . . . . . . . 8  |-  ( X  =  ( q (
join `  K )
r )  ->  ( P C X  <->  P C
( q ( join `  K ) r ) ) )
2423biimpd 207 . . . . . . 7  |-  ( X  =  ( q (
join `  K )
r )  ->  ( P C X  ->  P C ( q (
join `  K )
r ) ) )
259adantr 463 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  /\  P C ( q (
join `  K )
r ) )  ->  K  e.  HL )
26 simpll3 1035 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  P  e.  B )
2726, 12, 163jca 1174 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  ( P  e.  B  /\  q  e.  A  /\  r  e.  A )
)
2827adantr 463 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  /\  P C ( q (
join `  K )
r ) )  -> 
( P  e.  B  /\  q  e.  A  /\  r  e.  A
) )
29 simplr 753 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  /\  P C ( q (
join `  K )
r ) )  -> 
q  =/=  r )
30 simpr 459 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  /\  P C ( q (
join `  K )
r ) )  ->  P C ( q (
join `  K )
r ) )
31 lncvrelat.c . . . . . . . . . 10  |-  C  =  (  <o  `  K )
3213, 3, 31, 4cvrat2 35605 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  B  /\  q  e.  A  /\  r  e.  A
)  /\  ( q  =/=  r  /\  P C ( q ( join `  K ) r ) ) )  ->  P  e.  A )
3325, 28, 29, 30, 32syl112anc 1230 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  /\  P C ( q (
join `  K )
r ) )  ->  P  e.  A )
3433ex 432 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  ( P C ( q (
join `  K )
r )  ->  P  e.  A ) )
3524, 34syl9r 72 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  ( X  =  ( q
( join `  K )
r )  ->  ( P C X  ->  P  e.  A ) ) )
3622, 35sylbid 215 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  (
( M `  X
)  =  ( M `
 ( q (
join `  K )
r ) )  -> 
( P C X  ->  P  e.  A
) ) )
3736expimpd 601 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  ( q  e.  A  /\  r  e.  A
) )  ->  (
( q  =/=  r  /\  ( M `  X
)  =  ( M `
 ( q (
join `  K )
r ) ) )  ->  ( P C X  ->  P  e.  A ) ) )
3837rexlimdvva 2895 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  ->  ( E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  ( M `
 X )  =  ( M `  (
q ( join `  K
) r ) ) )  ->  ( P C X  ->  P  e.  A ) ) )
398, 38sylbid 215 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  ->  ( ( M `  X )  e.  N  ->  ( P C X  ->  P  e.  A
) ) )
4039imp32 431 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  ( ( M `  X )  e.  N  /\  P C X ) )  ->  P  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836    =/= wne 2591   E.wrex 2747   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   Basecbs 14657   joincjn 15713   Latclat 15815    <o ccvr 35439   Atomscatm 35440   HLchlt 35527   Linesclines 35670   pmapcpmap 35673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-preset 15697  df-poset 15715  df-plt 15728  df-lub 15744  df-glb 15745  df-join 15746  df-meet 15747  df-p0 15809  df-lat 15816  df-clat 15878  df-oposet 35353  df-ol 35355  df-oml 35356  df-covers 35443  df-ats 35444  df-atl 35475  df-cvlat 35499  df-hlat 35528  df-lines 35677  df-pmap 35680
This theorem is referenced by: (None)
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