Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ltrnset Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator