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Theorem isltrn 34423
Description: The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
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
isltrn ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((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 isltrn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ltrnset.l . . . 4 = (le‘𝐾)
2 ltrnset.j . . . 4 = (join‘𝐾)
3 ltrnset.m . . . 4 = (meet‘𝐾)
4 ltrnset.a . . . 4 𝐴 = (Atoms‘𝐾)
5 ltrnset.h . . . 4 𝐻 = (LHyp‘𝐾)
6 ltrnset.d . . . 4 𝐷 = ((LDil‘𝐾)‘𝑊)
7 ltrnset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7ltrnset 34422 . . 3 ((𝐾𝐵𝑊𝐻) → 𝑇 = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
98eleq2d 2673 . 2 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇𝐹 ∈ {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))}))
10 fveq1 6102 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
1110oveq2d 6565 . . . . . . 7 (𝑓 = 𝐹 → (𝑝 (𝑓𝑝)) = (𝑝 (𝐹𝑝)))
1211oveq1d 6564 . . . . . 6 (𝑓 = 𝐹 → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑝 (𝐹𝑝)) 𝑊))
13 fveq1 6102 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑞) = (𝐹𝑞))
1413oveq2d 6565 . . . . . . 7 (𝑓 = 𝐹 → (𝑞 (𝑓𝑞)) = (𝑞 (𝐹𝑞)))
1514oveq1d 6564 . . . . . 6 (𝑓 = 𝐹 → ((𝑞 (𝑓𝑞)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
1612, 15eqeq12d 2625 . . . . 5 (𝑓 = 𝐹 → (((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊) ↔ ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))
1716imbi2d 329 . . . 4 (𝑓 = 𝐹 → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
18172ralbidv 2972 . . 3 (𝑓 = 𝐹 → (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
1918elrab 3331 . 2 (𝐹 ∈ {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))} ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))))
209, 19syl6bb 275 1 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇 ↔ (𝐹𝐷 ∧ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝐹𝑝)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900   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:  isltrn2N  34424  ltrnu  34425  ltrnldil  34426  ltrncnv  34450  idltrn  34454  cdleme50ltrn  34863  ltrnco  35025
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