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Theorem isltrn2N 34424
 Description: The predicate "is a lattice translation". Version of isltrn 34423 that considers only different 𝑝 and 𝑞. TODO: Can this eliminate some separate proofs for the 𝑝 = 𝑞 case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ltrnset.l = (le‘𝐾)
ltrnset.j = (join‘𝐾)
ltrnset.m = (meet‘𝐾)
ltrnset.a 𝐴 = (Atoms‘𝐾)
ltrnset.h 𝐻 = (LHyp‘𝐾)
ltrnset.d 𝐷 = ((LDil‘𝐾)‘𝑊)
ltrnset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
isltrn2N ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
Distinct variable groups:   𝑞,𝑝,𝐴   𝐾,𝑝,𝑞   𝑊,𝑝,𝑞   𝐹,𝑝,𝑞
Allowed substitution hints:   𝐵(𝑞,𝑝)   𝐷(𝑞,𝑝)   𝑇(𝑞,𝑝)   𝐻(𝑞,𝑝)   (𝑞,𝑝)   (𝑞,𝑝)   (𝑞,𝑝)

Proof of Theorem isltrn2N
StepHypRef Expression
1 ltrnset.l . . 3 = (le‘𝐾)
2 ltrnset.j . . 3 = (join‘𝐾)
3 ltrnset.m . . 3 = (meet‘𝐾)
4 ltrnset.a . . 3 𝐴 = (Atoms‘𝐾)
5 ltrnset.h . . 3 𝐻 = (LHyp‘𝐾)
6 ltrnset.d . . 3 𝐷 = ((LDil‘𝐾)‘𝑊)
7 ltrnset.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7isltrn 34423 . 2 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
9 3simpa 1051 . . . . . 6 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊))
109imim1i 61 . . . . 5 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
11 3anass 1035 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
12 3anrot 1036 . . . . . . . . 9 ((𝑝𝑞 ∧ ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞))
13 df-ne 2782 . . . . . . . . . 10 (𝑝𝑞 ↔ ¬ 𝑝 = 𝑞)
1413anbi1i 727 . . . . . . . . 9 ((𝑝𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1511, 12, 143bitr3i 289 . . . . . . . 8 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
1615imbi1i 338 . . . . . . 7 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
17 impexp 461 . . . . . . 7 (((¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ (¬ 𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
1816, 17bitri 263 . . . . . 6 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ (¬ 𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
19 id 22 . . . . . . . . . 10 (𝑝 = 𝑞𝑝 = 𝑞)
20 fveq2 6103 . . . . . . . . . 10 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
2119, 20oveq12d 6567 . . . . . . . . 9 (𝑝 = 𝑞 → (𝑝 (𝐹𝑝)) = (𝑞 (𝐹𝑞)))
2221oveq1d 6564 . . . . . . . 8 (𝑝 = 𝑞 → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
2322a1d 25 . . . . . . 7 (𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
24 pm2.61 182 . . . . . . 7 ((𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) → ((¬ 𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
2523, 24ax-mp 5 . . . . . 6 ((¬ 𝑝 = 𝑞 → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2618, 25sylbi 206 . . . . 5 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2710, 26impbii 198 . . . 4 (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
28272ralbii 2964 . . 3 (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
2928anbi2i 726 . 2 ((𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))) ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
308, 29syl6bb 275 1 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝑝𝑞) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  LHypclh 34288  LDilcldil 34404  LTrncltrn 34405 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-ltrn 34409 This theorem is referenced by: (None)
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