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Theorem lncvrelatN 34452
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 34035 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1012 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  ->  K  e.  Lat )
3 eqid 2460 . . . . 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 34445 . . . 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 1030 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  K  e.  HL )
10 simpll2 1031 . . . . . . 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 33961 . . . . . . . . 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 33961 . . . . . . . . 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 15527 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  q  e.  B  /\  r  e.  B )  ->  ( q ( join `  K ) r )  e.  B )
2011, 15, 18, 19syl3anc 1223 . . . . . . 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 34433 . . . . . . 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 1223 . . . . . 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 4444 . . . . . . . 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 465 . . . . . . . . 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 1032 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  P  e.  B )
2726, 12, 163jca 1171 . . . . . . . . . 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 465 . . . . . . . . 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 754 . . . . . . . . 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 461 . . . . . . . . 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 34100 . . . . . . . . 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 1227 . . . . . . . 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 434 . . . . . . 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 603 . . . 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 2955 . . 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 433 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   joincjn 15420   Latclat 15521    <o ccvr 33934   Atomscatm 33935   HLchlt 34022   Linesclines 34165   pmapcpmap 34168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-lines 34172  df-pmap 34175
This theorem is referenced by: (None)
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