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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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