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

Theorem isline3 30635
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 30223 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
21adantr 453 . . 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 30633 . . 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 16 . 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 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  ->  K  e.  HL )
10 simplr 733 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  ->  X  e.  B )
111ad2antrr 708 . . . . . 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 30149 . . . . . . 7  |-  ( p  e.  A  ->  p  e.  B )
1413ad2antrl 710 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  ->  p  e.  B )
1512, 4atbase 30149 . . . . . . 7  |-  ( q  e.  A  ->  q  e.  B )
1615ad2antll 711 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
q  e.  B )
1712, 3latjcl 14481 . . . . . 6  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  q  e.  B )  ->  ( p  .\/  q
)  e.  B )
1811, 14, 16, 17syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
( p  .\/  q
)  e.  B )
1912, 6pmap11 30621 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  ( p  .\/  q )  e.  B )  -> 
( ( M `  X )  =  ( M `  ( p 
.\/  q ) )  <-> 
X  =  ( p 
.\/  q ) ) )
209, 10, 18, 19syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( p  e.  A  /\  q  e.  A ) )  -> 
( ( M `  X )  =  ( M `  ( p 
.\/  q ) )  <-> 
X  =  ( p 
.\/  q ) ) )
2120anbi2d 686 . . 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 2748 . 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 246 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   ` cfv 5456  (class class class)co 6083   Basecbs 13471   joincjn 14403   Latclat 14476   Atomscatm 30123   HLchlt 30210   Linesclines 30353   pmapcpmap 30356
This theorem is referenced by:  isline4N  30636  lneq2at  30637  lnatexN  30638  lncvrat  30641  lncmp  30642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lines 30360  df-pmap 30363
  Copyright terms: Public domain W3C validator