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Theorem isline4N 29966
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 2283 . . 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 29965 . 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 29479 . . . . . 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 2283 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 isline4.c . . . . . 6  |-  C  =  (  <o  `  K )
131, 11, 2, 12, 3cvrval3 29602 . . . . 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 1182 . . . 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 29550 . . . . . . . . 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 29502 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  p  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
2016, 17, 18, 19syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A )  /\  q  e.  A )  ->  ( -.  q ( le `  K ) p  <->  q  =/=  p ) )
21 necom 2527 . . . . . . 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 2285 . . . . . . 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 2560 . . . 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 2560 . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078    <o ccvr 29452   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   Linesclines 29683   pmapcpmap 29686
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
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