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

Theorem trnfsetN 35963
Description: The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
trnset.a 𝐴 = (Atoms‘𝐾)
trnset.s 𝑆 = (PSubSp‘𝐾)
trnset.p + = (+𝑃𝐾)
trnset.o = (⊥𝑃𝐾)
trnset.w 𝑊 = (WAtoms‘𝐾)
trnset.m 𝑀 = (PAut‘𝐾)
trnset.l 𝐿 = (Dil‘𝐾)
trnset.t 𝑇 = (Trn‘𝐾)
Assertion
Ref Expression
trnfsetN (𝐾𝐶𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
Distinct variable groups:   𝐴,𝑑   𝑓,𝑑,𝑞,𝑟,𝐾   𝑓,𝐿   𝑊,𝑞,𝑟
Allowed substitution hints:   𝐴(𝑓,𝑟,𝑞)   𝐶(𝑓,𝑟,𝑞,𝑑)   + (𝑓,𝑟,𝑞,𝑑)   𝑆(𝑓,𝑟,𝑞,𝑑)   𝑇(𝑓,𝑟,𝑞,𝑑)   𝐿(𝑟,𝑞,𝑑)   𝑀(𝑓,𝑟,𝑞,𝑑)   (𝑓,𝑟,𝑞,𝑑)   𝑊(𝑓,𝑑)

Proof of Theorem trnfsetN
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3352 . 2 (𝐾𝐶𝐾 ∈ V)
2 trnset.t . . 3 𝑇 = (Trn‘𝐾)
3 fveq2 6353 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 trnset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2812 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6353 . . . . . . . 8 (𝑘 = 𝐾 → (Dil‘𝑘) = (Dil‘𝐾))
7 trnset.l . . . . . . . 8 𝐿 = (Dil‘𝐾)
86, 7syl6eqr 2812 . . . . . . 7 (𝑘 = 𝐾 → (Dil‘𝑘) = 𝐿)
98fveq1d 6355 . . . . . 6 (𝑘 = 𝐾 → ((Dil‘𝑘)‘𝑑) = (𝐿𝑑))
10 fveq2 6353 . . . . . . . . 9 (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
11 trnset.w . . . . . . . . 9 𝑊 = (WAtoms‘𝐾)
1210, 11syl6eqr 2812 . . . . . . . 8 (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
1312fveq1d 6355 . . . . . . 7 (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊𝑑))
14 fveq2 6353 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (+𝑃𝑘) = (+𝑃𝐾))
15 trnset.p . . . . . . . . . . . 12 + = (+𝑃𝐾)
1614, 15syl6eqr 2812 . . . . . . . . . . 11 (𝑘 = 𝐾 → (+𝑃𝑘) = + )
1716oveqd 6831 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(+𝑃𝑘)(𝑓𝑞)) = (𝑞 + (𝑓𝑞)))
18 fveq2 6353 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
19 trnset.o . . . . . . . . . . . 12 = (⊥𝑃𝐾)
2018, 19syl6eqr 2812 . . . . . . . . . . 11 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
2120fveq1d 6355 . . . . . . . . . 10 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ( ‘{𝑑}))
2217, 21ineq12d 3958 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})))
2316oveqd 6831 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(+𝑃𝑘)(𝑓𝑟)) = (𝑟 + (𝑓𝑟)))
2423, 21ineq12d 3958 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})))
2522, 24eqeq12d 2775 . . . . . . . 8 (𝑘 = 𝐾 → (((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
2613, 25raleqbidv 3291 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
2713, 26raleqbidv 3291 . . . . . 6 (𝑘 = 𝐾 → (∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
289, 27rabeqbidv 3335 . . . . 5 (𝑘 = 𝐾 → {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑}))} = {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))})
295, 28mpteq12dv 4885 . . . 4 (𝑘 = 𝐾 → (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑}))}) = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
30 df-trnN 35914 . . . 4 Trn = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑}))}))
31 fvex 6363 . . . . . 6 (Atoms‘𝐾) ∈ V
324, 31eqeltri 2835 . . . . 5 𝐴 ∈ V
3332mptex 6651 . . . 4 (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}) ∈ V
3429, 30, 33fvmpt 6445 . . 3 (𝐾 ∈ V → (Trn‘𝐾) = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
352, 34syl5eq 2806 . 2 (𝐾 ∈ V → 𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
361, 35syl 17 1 (𝐾𝐶𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  cin 3714  {csn 4321  cmpt 4881  cfv 6049  (class class class)co 6814  Atomscatm 35071  PSubSpcpsubsp 35303  +𝑃cpadd 35602  𝑃cpolN 35709  WAtomscwpointsN 35793  PAutcpautN 35794  DilcdilN 35909  TrnctrnN 35910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-trnN 35914
This theorem is referenced by:  trnsetN  35964
  Copyright terms: Public domain W3C validator