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Theorem isline4N 30574
Description: Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
isline4.b  |-  B  =  ( Base `  K
)
isline4.c  |-  C  =  (  <o  `  K )
isline4.a  |-  A  =  ( Atoms `  K )
isline4.n  |-  N  =  ( Lines `  K )
isline4.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
isline4N  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  p C X ) )
Distinct variable groups:    A, p    B, p    K, p    M, p    X, p
Allowed substitution hints:    C( p)    N( p)

Proof of Theorem isline4N
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 isline4.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2436 . . 3  |-  ( join `  K )  =  (
join `  K )
3 isline4.a . . 3  |-  A  =  ( Atoms `  K )
4 isline4.n . . 3  |-  N  =  ( Lines `  K )
5 isline4.m . . 3  |-  M  =  ( pmap `  K
)
61, 2, 3, 4, 5isline3 30573 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  E. q  e.  A  (
p  =/=  q  /\  X  =  ( p
( join `  K )
q ) ) ) )
7 simpll 731 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
81, 3atbase 30087 . . . . . 6  |-  ( p  e.  A  ->  p  e.  B )
98adantl 453 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  p  e.  B )
10 simplr 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  X  e.  B )
11 eqid 2436 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 isline4.c . . . . . 6  |-  C  =  (  <o  `  K )
131, 11, 2, 12, 3cvrval3 30210 . . . . 5  |-  ( ( K  e.  HL  /\  p  e.  B  /\  X  e.  B )  ->  ( p C X  <->  E. q  e.  A  ( -.  q ( le `  K ) p  /\  ( p (
join `  K )
q )  =  X ) ) )
147, 9, 10, 13syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( p C X  <->  E. q  e.  A  ( -.  q ( le `  K ) p  /\  ( p (
join `  K )
q )  =  X ) ) )
15 hlatl 30158 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
1615ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  K  e.  AtLat )
17 simpr 448 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  q  e.  A )
18 simplr 732 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  p  e.  A )
1911, 3atncmp 30110 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  p  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
2016, 17, 18, 19syl3anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
21 necom 2685 . . . . . . 7  |-  ( q  =/=  p  <->  p  =/=  q )
2220, 21syl6bb 253 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  p  =/=  q ) )
23 eqcom 2438 . . . . . . 7  |-  ( ( p ( join `  K
) q )  =  X  <->  X  =  (
p ( join `  K
) q ) )
2423a1i 11 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  (
( p ( join `  K ) q )  =  X  <->  X  =  ( p ( join `  K ) q ) ) )
2522, 24anbi12d 692 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  (
( -.  q ( le `  K ) p  /\  ( p ( join `  K
) q )  =  X )  <->  ( p  =/=  q  /\  X  =  ( p ( join `  K ) q ) ) ) )
2625rexbidva 2722 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( E. q  e.  A  ( -.  q ( le `  K ) p  /\  ( p ( join `  K ) q )  =  X )  <->  E. q  e.  A  ( p  =/=  q  /\  X  =  ( p ( join `  K ) q ) ) ) )
2714, 26bitrd 245 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( p C X  <->  E. q  e.  A  ( p  =/=  q  /\  X  =  (
p ( join `  K
) q ) ) ) )
2827rexbidva 2722 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( E. p  e.  A  p C X  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  (
p ( join `  K
) q ) ) ) )
296, 28bitr4d 248 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  p C X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401    <o ccvr 30060   Atomscatm 30061   AtLatcal 30062   HLchlt 30148   Linesclines 30291   pmapcpmap 30294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lines 30298  df-pmap 30301
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