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Theorem ltrnset 37407
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 37406 . . . 4 (𝐾𝐵 → (LTrn‘𝐾) = (𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))}))
87fveq1d 6651 . . 3 (𝐾𝐵 → ((LTrn‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})‘𝑊))
91, 8syl5eq 2848 . 2 (𝐾𝐵𝑇 = ((𝑤𝐻 ↦ {𝑓 ∈ ((LDil‘𝐾)‘𝑤) ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤))})‘𝑊))
10 fveq2 6649 . . . . 5 (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = ((LDil‘𝐾)‘𝑊))
11 ltrnset.d . . . . 5 𝐷 = ((LDil‘𝐾)‘𝑊)
1210, 11eqtr4di 2854 . . . 4 (𝑤 = 𝑊 → ((LDil‘𝐾)‘𝑤) = 𝐷)
13 breq2 5037 . . . . . . . 8 (𝑤 = 𝑊 → (𝑝 𝑤𝑝 𝑊))
1413notbid 321 . . . . . . 7 (𝑤 = 𝑊 → (¬ 𝑝 𝑤 ↔ ¬ 𝑝 𝑊))
15 breq2 5037 . . . . . . . 8 (𝑤 = 𝑊 → (𝑞 𝑤𝑞 𝑊))
1615notbid 321 . . . . . . 7 (𝑤 = 𝑊 → (¬ 𝑞 𝑤 ↔ ¬ 𝑞 𝑊))
1714, 16anbi12d 633 . . . . . 6 (𝑤 = 𝑊 → ((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)))
18 oveq2 7147 . . . . . . 7 (𝑤 = 𝑊 → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑝 (𝑓𝑝)) 𝑊))
19 oveq2 7147 . . . . . . 7 (𝑤 = 𝑊 → ((𝑞 (𝑓𝑞)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑊))
2018, 19eqeq12d 2817 . . . . . 6 (𝑤 = 𝑊 → (((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤) ↔ ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊)))
2117, 20imbi12d 348 . . . . 5 (𝑤 = 𝑊 → (((¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤) → ((𝑝 (𝑓𝑝)) 𝑤) = ((𝑞 (𝑓𝑞)) 𝑤)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))))
22212ralbidv 3167 . . . 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 3436 . . 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 2801 . . 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))})
2511fvexi 6663 . . . 4 𝐷 ∈ V
2625rabex 5202 . . 3 {𝑓𝐷 ∣ ∀𝑝𝐴𝑞𝐴 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) → ((𝑝 (𝑓𝑝)) 𝑊) = ((𝑞 (𝑓𝑞)) 𝑊))} ∈ V
2723, 24, 26fvmpt 6749 . 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))})
289, 27sylan9eq 2856 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 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113   class class class wbr 5033  cmpt 5113  cfv 6328  (class class class)co 7139  lecple 16567  joincjn 17549  meetcmee 17550  Atomscatm 36552  LHypclh 37273  LDilcldil 37389  LTrncltrn 37390
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-ltrn 37394
This theorem is referenced by:  isltrn  37408
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