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Theorem isline4N 30259
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 2404 . . 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 30258 . 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 29772 . . . . . 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 2404 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 isline4.c . . . . . 6  |-  C  =  (  <o  `  K )
131, 11, 2, 12, 3cvrval3 29895 . . . . 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 29843 . . . . . . . . 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 29795 . . . . . . . 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 2648 . . . . . . 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 2406 . . . . . . 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 2683 . . . 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 2683 . 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 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356    <o ccvr 29745   Atomscatm 29746   AtLatcal 29747   HLchlt 29833   Linesclines 29976   pmapcpmap 29979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lines 29983  df-pmap 29986
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