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Theorem isline4N 30037
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 2366 . . 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 30036 . 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 730 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  K  e.  HL )
81, 3atbase 29550 . . . . . 6  |-  ( p  e.  A  ->  p  e.  B )
98adantl 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  p  e.  B )
10 simplr 731 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  X  e.  B )
11 eqid 2366 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 isline4.c . . . . . 6  |-  C  =  (  <o  `  K )
131, 11, 2, 12, 3cvrval3 29673 . . . . 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 1183 . . . 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 29621 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
1615ad3antrrr 710 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  K  e.  AtLat )
17 simpr 447 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  q  e.  A )
18 simplr 731 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  p  e.  A )
1911, 3atncmp 29573 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  p  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
2016, 17, 18, 19syl3anc 1183 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
21 necom 2610 . . . . . . 7  |-  ( q  =/=  p  <->  p  =/=  q )
2220, 21syl6bb 252 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  p  =/=  q ) )
23 eqcom 2368 . . . . . . 7  |-  ( ( p ( join `  K
) q )  =  X  <->  X  =  (
p ( join `  K
) q ) )
2423a1i 10 . . . . . 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 691 . . . . 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 2645 . . . 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 244 . . 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 2645 . 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 247 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 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288    <o ccvr 29523   Atomscatm 29524   AtLatcal 29525   HLchlt 29611   Linesclines 29754   pmapcpmap 29757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-lines 29761  df-pmap 29764
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