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Theorem lncvrelatN 29970
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 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  ->  K  e.  Lat )
3 eqid 2283 . . . . 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 29963 . . . 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 15 . . 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 994 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  K  e.  HL )
10 simpll2 995 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  X  e.  B )
119, 1syl 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  K  e.  Lat )
12 simplrl 736 . . . . . . . . 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 29479 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
1512, 14syl 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  q  e.  B )
16 simplrr 737 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  r  e.  A )
1713, 4atbase 29479 . . . . . . . . 9  |-  ( r  e.  A  ->  r  e.  B )
1816, 17syl 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  r  e.  B )
1913, 3latjcl 14156 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  q  e.  B  /\  r  e.  B )  ->  ( q ( join `  K ) r )  e.  B )
2011, 15, 18, 19syl3anc 1182 . . . . . . 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 29951 . . . . . . 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 1182 . . . . . 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 4027 . . . . . . . 8  |-  ( X  =  ( q (
join `  K )
r )  ->  ( P C X  <->  P C
( q ( join `  K ) r ) ) )
2423biimpd 198 . . . . . . 7  |-  ( X  =  ( q (
join `  K )
r )  ->  ( P C X  ->  P C ( q (
join `  K )
r ) ) )
259adantr 451 . . . . . . . . 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 996 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  (
q  e.  A  /\  r  e.  A )
)  /\  q  =/=  r )  ->  P  e.  B )
2726, 12, 163jca 1132 . . . . . . . . . 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 451 . . . . . . . . 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 731 . . . . . . . . 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 447 . . . . . . . . 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 29618 . . . . . . . . 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 1186 . . . . . . . 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 423 . . . . . . 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 67 . . . . . 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 206 . . . . 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 586 . . . 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 2674 . . 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 206 . 2  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  ->  ( ( M `  X )  e.  N  ->  ( P C X  ->  P  e.  A
) ) )
4039imp32 422 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  /\  ( ( M `  X )  e.  N  /\  P C X ) )  ->  P  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   Latclat 14151    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   Linesclines 29683   pmapcpmap 29686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lines 29690  df-pmap 29693
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