Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isline3 Unicode version

Theorem isline3 29965
Description: Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)
Hypotheses
Ref Expression
isline3.b  |-  B  =  ( Base `  K
)
isline3.j  |-  .\/  =  ( join `  K )
isline3.a  |-  A  =  ( Atoms `  K )
isline3.n  |-  N  =  ( Lines `  K )
isline3.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
isline3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  E. q  e.  A  (
p  =/=  q  /\  X  =  ( p  .\/  q ) ) ) )
Distinct variable groups:    q, p, B    A, p, q    K, p, q    M, p, q    X, p, q
Allowed substitution hints:    .\/ ( q, p)    N( q, p)

Proof of Theorem isline3
StepHypRef Expression
1 hllat 29553 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
21adantr 451 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  K  e.  Lat )
3 isline3.j . . . 4  |-  .\/  =  ( join `  K )
4 isline3.a . . . 4  |-  A  =  ( Atoms `  K )
5 isline3.n . . . 4  |-  N  =  ( Lines `  K )
6 isline3.m . . . 4  |-  M  =  ( pmap `  K
)
73, 4, 5, 6isline2 29963 . . 3  |-  ( K  e.  Lat  ->  (
( M `  X
)  e.  N  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  ( M `  X )  =  ( M `  ( p  .\/  q ) ) ) ) )
82, 7syl 15 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  E. q  e.  A  (
p  =/=  q  /\  ( M `  X )  =  ( M `  ( p  .\/  q ) ) ) ) )
9 simpll 730 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  ->  K  e.  HL )
10 simplr 731 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  ->  X  e.  B )
111ad2antrr 706 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  ->  K  e.  Lat )
12 isline3.b . . . . . . . 8  |-  B  =  ( Base `  K
)
1312, 4atbase 29479 . . . . . . 7  |-  ( p  e.  A  ->  p  e.  B )
1413ad2antrl 708 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  ->  p  e.  B )
1512, 4atbase 29479 . . . . . . 7  |-  ( q  e.  A  ->  q  e.  B )
1615ad2antll 709 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
q  e.  B )
1712, 3latjcl 14156 . . . . . 6  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  q  e.  B )  ->  ( p  .\/  q
)  e.  B )
1811, 14, 16, 17syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
( p  .\/  q
)  e.  B )
1912, 6pmap11 29951 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( p  .\/  q )  e.  B )  -> 
( ( M `  X )  =  ( M `  ( p 
.\/  q ) )  <-> 
X  =  ( p 
.\/  q ) ) )
209, 10, 18, 19syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
( ( M `  X )  =  ( M `  ( p 
.\/  q ) )  <-> 
X  =  ( p 
.\/  q ) ) )
2120anbi2d 684 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
( ( p  =/=  q  /\  ( M `
 X )  =  ( M `  (
p  .\/  q )
) )  <->  ( p  =/=  q  /\  X  =  ( p  .\/  q
) ) ) )
22212rexbidva 2584 . 2  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  ( M `
 X )  =  ( M `  (
p  .\/  q )
) )  <->  E. p  e.  A  E. q  e.  A  ( p  =/=  q  /\  X  =  ( p  .\/  q
) ) ) )
238, 22bitrd 244 1  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( ( M `  X )  e.  N  <->  E. p  e.  A  E. q  e.  A  (
p  =/=  q  /\  X  =  ( p  .\/  q ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540   Linesclines 29683   pmapcpmap 29686
This theorem is referenced by:  isline4N  29966  lneq2at  29967  lnatexN  29968  lncvrat  29971  lncmp  29972
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
  Copyright terms: Public domain W3C validator