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Theorem lncvrelatN 30592
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 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  B )  ->  K  e.  Lat )
3 eqid 2296 . . . . 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 30585 . . . 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 30101 . . . . . . . . 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 30101 . . . . . . . . 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 14172 . . . . . . . 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 30573 . . . . . . 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 4043 . . . . . . . 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 30240 . . . . . . . . 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 2687 . . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   Latclat 14167    <o ccvr 30074   Atomscatm 30075   HLchlt 30162   Linesclines 30305   pmapcpmap 30308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lines 30312  df-pmap 30315
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