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Theorem isline4N 37019
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 = ( ⋖ ‘𝐾)
isline4.a 𝐴 = (Atoms‘𝐾)
isline4.n 𝑁 = (Lines‘𝐾)
isline4.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
isline4N ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
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 2824 . . 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 37018 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
7 simpll 766 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
81, 3atbase 36531 . . . . . 6 (𝑝𝐴𝑝𝐵)
98adantl 485 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
10 simplr 768 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
11 eqid 2824 . . . . . 6 (le‘𝐾) = (le‘𝐾)
12 isline4.c . . . . . 6 𝐶 = ( ⋖ ‘𝐾)
131, 11, 2, 12, 3cvrval3 36655 . . . . 5 ((𝐾 ∈ HL ∧ 𝑝𝐵𝑋𝐵) → (𝑝𝐶𝑋 ↔ ∃𝑞𝐴𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋)))
147, 9, 10, 13syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → (𝑝𝐶𝑋 ↔ ∃𝑞𝐴𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋)))
15 hlatl 36602 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
1615ad3antrrr 729 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → 𝐾 ∈ AtLat)
17 simpr 488 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → 𝑞𝐴)
18 simplr 768 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → 𝑝𝐴)
1911, 3atncmp 36554 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑞𝐴𝑝𝐴) → (¬ 𝑞(le‘𝐾)𝑝𝑞𝑝))
2016, 17, 18, 19syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → (¬ 𝑞(le‘𝐾)𝑝𝑞𝑝))
21 necom 3067 . . . . . . 7 (𝑞𝑝𝑝𝑞)
2220, 21syl6bb 290 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → (¬ 𝑞(le‘𝐾)𝑝𝑝𝑞))
23 eqcom 2831 . . . . . . 7 ((𝑝(join‘𝐾)𝑞) = 𝑋𝑋 = (𝑝(join‘𝐾)𝑞))
2423a1i 11 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → ((𝑝(join‘𝐾)𝑞) = 𝑋𝑋 = (𝑝(join‘𝐾)𝑞)))
2522, 24anbi12d 633 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → ((¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋) ↔ (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
2625rexbidva 3289 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → (∃𝑞𝐴𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋) ↔ ∃𝑞𝐴 (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
2714, 26bitrd 282 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ 𝑝𝐴) → (𝑝𝐶𝑋 ↔ ∃𝑞𝐴 (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
2827rexbidva 3289 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (∃𝑝𝐴 𝑝𝐶𝑋 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑝(join‘𝐾)𝑞))))
296, 28bitr4d 285 1 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wrex 3134   class class class wbr 5053  cfv 6344  (class class class)co 7150  Basecbs 16486  lecple 16575  joincjn 17557  ccvr 36504  Atomscatm 36505  AtLatcal 36506  HLchlt 36592  Linesclines 36736  pmapcpmap 36739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-lat 17659  df-clat 17721  df-oposet 36418  df-ol 36420  df-oml 36421  df-covers 36508  df-ats 36509  df-atl 36540  df-cvlat 36564  df-hlat 36593  df-lines 36743  df-pmap 36746
This theorem is referenced by: (None)
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