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Theorem isltrn 35920
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 35919 . . 3 ((𝐾𝐵𝑊𝐻) → 𝑇 = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
98eleq2d 2835 . 2 ((𝐾𝐵𝑊𝐻) → (𝐹𝑇𝐹 ∈ {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))}))
10 fveq1 6331 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
1110oveq2d 6808 . . . . . . 7 (𝑓 = 𝐹 → (𝑝 (𝑓𝑝)) = (𝑝 (𝐹𝑝)))
1211oveq1d 6807 . . . . . 6 (𝑓 = 𝐹 → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑝 (𝐹𝑝)) 𝑊))
13 fveq1 6331 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑞) = (𝐹𝑞))
1413oveq2d 6808 . . . . . . 7 (𝑓 = 𝐹 → (𝑞 (𝑓𝑞)) = (𝑞 (𝐹𝑞)))
1514oveq1d 6807 . . . . . 6 (𝑓 = 𝐹 → ((𝑞 (𝑓𝑞)) 𝑊) = ((𝑞 (𝐹𝑞)) 𝑊))
1612, 15eqeq12d 2785 . . . . 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 3137 . . 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 3513 . 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 276 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 196  wa 382   = wceq 1630  wcel 2144  wral 3060  {crab 3064   class class class wbr 4784  cfv 6031  (class class class)co 6792  lecple 16155  joincjn 17151  meetcmee 17152  Atomscatm 35065  LHypclh 35785  LDilcldil 35901  LTrncltrn 35902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-ltrn 35906
This theorem is referenced by:  isltrn2N  35921  ltrnu  35922  ltrnldil  35923  ltrncnv  35947  idltrn  35951  cdleme50ltrn  36359  ltrnco  36521
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