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Theorem ltrnset 35922
Description: The set of lattice translations for a fiducial co-atom 𝑊. (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
ltrnset ((𝐾𝐵𝑊𝐻) → 𝑇 = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
Distinct variable groups:   𝑞,𝑝,𝐴   𝐷,𝑓   𝑓,𝑝,𝑞,𝐾   𝑓,𝑊,𝑝,𝑞
Allowed substitution hints:   𝐴(𝑓)   𝐵(𝑓,𝑞,𝑝)   𝐷(𝑞,𝑝)   𝑇(𝑓,𝑞,𝑝)   𝐻(𝑓,𝑞,𝑝)   (𝑓,𝑞,𝑝)   (𝑓,𝑞,𝑝)   (𝑓,𝑞,𝑝)

Proof of Theorem ltrnset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ltrnset.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
2 ltrnset.l . . . . 5 = (le‘𝐾)
3 ltrnset.j . . . . 5 = (join‘𝐾)
4 ltrnset.m . . . . 5 = (meet‘𝐾)
5 ltrnset.a . . . . 5 𝐴 = (Atoms‘𝐾)
6 ltrnset.h . . . . 5 𝐻 = (LHyp‘𝐾)
72, 3, 4, 5, 6ltrnfset 35921 . . . 4 (𝐾𝐵 → (LTrn‘𝐾) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}))
87fveq1d 6334 . . 3 (𝐾𝐵 → ((LTrn‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})‘𝑊))
91, 8syl5eq 2817 . 2 (𝐾𝐵𝑇 = ((𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})‘𝑊))
10 fveq2 6332 . . . . 5 (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = ((LDil‘𝐾)‘𝑊))
11 ltrnset.d . . . . 5 𝐷 = ((LDil‘𝐾)‘𝑊)
1210, 11syl6eqr 2823 . . . 4 (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = 𝐷)
13 breq2 4790 . . . . . . . 8 (𝑤 = 𝑊 → (𝑝 𝑤𝑝 𝑊))
1413notbid 307 . . . . . . 7 (𝑤 = 𝑊 → (¬ 𝑝 𝑤 ↔ ¬ 𝑝 𝑊))
15 breq2 4790 . . . . . . . 8 (𝑤 = 𝑊 → (𝑞 𝑤𝑞 𝑊))
1615notbid 307 . . . . . . 7 (𝑤 = 𝑊 → (¬ 𝑞 𝑤 ↔ ¬ 𝑞 𝑊))
1714, 16anbi12d 608 . . . . . 6 (𝑤 = 𝑊 → ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
18 oveq2 6800 . . . . . . 7 (𝑤 = 𝑊 → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑝 (𝑓𝑝)) 𝑊))
19 oveq2 6800 . . . . . . 7 (𝑤 = 𝑊 → ((𝑞 (𝑓𝑞)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑊))
2018, 19eqeq12d 2786 . . . . . 6 (𝑤 = 𝑊 → (((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤) ↔ ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊)))
2117, 20imbi12d 333 . . . . 5 (𝑤 = 𝑊 → (((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))))
22212ralbidv 3138 . . . 4 (𝑤 = 𝑊 → (∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤)) ↔ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))))
2312, 22rabeqbidv 3345 . . 3 (𝑤 = 𝑊 → {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))} = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
24 eqid 2771 . . 3 (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})
25 fvex 6342 . . . . 5 ((LDil‘𝐾)‘𝑊) ∈ V
2611, 25eqeltri 2846 . . . 4 𝐷 ∈ V
2726rabex 4946 . . 3 {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))} ∈ V
2823, 24, 27fvmpt 6424 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})‘𝑊) = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
299, 28sylan9eq 2825 1 ((𝐾𝐵𝑊𝐻) → 𝑇 = {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351   class class class wbr 4786  cmpt 4863  cfv 6031  (class class class)co 6792  lecple 16155  joincjn 17151  meetcmee 17152  Atomscatm 35068  LHypclh 35788  LDilcldil 35904  LTrncltrn 35905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  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 35909
This theorem is referenced by:  isltrn  35923
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