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Theorem trnfsetN 34263
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 3184 . 2 (𝐾𝐶𝐾 ∈ V)
2 trnset.t . . 3 𝑇 = (Trn‘𝐾)
3 fveq2 6088 . . . . . 6 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4 trnset.a . . . . . 6 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2661 . . . . 5 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6 fveq2 6088 . . . . . . . 8 (𝑘 = 𝐾 → (Dil‘𝑘) = (Dil‘𝐾))
7 trnset.l . . . . . . . 8 𝐿 = (Dil‘𝐾)
86, 7syl6eqr 2661 . . . . . . 7 (𝑘 = 𝐾 → (Dil‘𝑘) = 𝐿)
98fveq1d 6090 . . . . . 6 (𝑘 = 𝐾 → ((Dil‘𝑘)‘𝑑) = (𝐿𝑑))
10 fveq2 6088 . . . . . . . . 9 (𝑘 = 𝐾 → (WAtoms‘𝑘) = (WAtoms‘𝐾))
11 trnset.w . . . . . . . . 9 𝑊 = (WAtoms‘𝐾)
1210, 11syl6eqr 2661 . . . . . . . 8 (𝑘 = 𝐾 → (WAtoms‘𝑘) = 𝑊)
1312fveq1d 6090 . . . . . . 7 (𝑘 = 𝐾 → ((WAtoms‘𝑘)‘𝑑) = (𝑊𝑑))
14 fveq2 6088 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (+𝑃𝑘) = (+𝑃𝐾))
15 trnset.p . . . . . . . . . . . 12 + = (+𝑃𝐾)
1614, 15syl6eqr 2661 . . . . . . . . . . 11 (𝑘 = 𝐾 → (+𝑃𝑘) = + )
1716oveqd 6544 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑞(+𝑃𝑘)(𝑓𝑞)) = (𝑞 + (𝑓𝑞)))
18 fveq2 6088 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (⊥𝑃𝑘) = (⊥𝑃𝐾))
19 trnset.o . . . . . . . . . . . 12 = (⊥𝑃𝐾)
2018, 19syl6eqr 2661 . . . . . . . . . . 11 (𝑘 = 𝐾 → (⊥𝑃𝑘) = )
2120fveq1d 6090 . . . . . . . . . 10 (𝑘 = 𝐾 → ((⊥𝑃𝑘)‘{𝑑}) = ( ‘{𝑑}))
2217, 21ineq12d 3776 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})))
2316oveqd 6544 . . . . . . . . . 10 (𝑘 = 𝐾 → (𝑟(+𝑃𝑘)(𝑓𝑟)) = (𝑟 + (𝑓𝑟)))
2423, 21ineq12d 3776 . . . . . . . . 9 (𝑘 = 𝐾 → ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑})))
2522, 24eqeq12d 2624 . . . . . . . 8 (𝑘 = 𝐾 → (((𝑞(+𝑃𝑘)(𝑓𝑞)) ∩ ((⊥𝑃𝑘)‘{𝑑})) = ((𝑟(+𝑃𝑘)(𝑓𝑟)) ∩ ((⊥𝑃𝑘)‘{𝑑})) ↔ ((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))))
2613, 25raleqbidv 3128 . . . . . . 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 3128 . . . . . 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 3167 . . . . 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 4657 . . . 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 34214 . . . 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 6098 . . . . . 6 (Atoms‘𝐾) ∈ V
324, 31eqeltri 2683 . . . . 5 𝐴 ∈ V
3332mptex 6368 . . . 4 (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}) ∈ V
3429, 30, 33fvmpt 6176 . . 3 (𝐾 ∈ V → (Trn‘𝐾) = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))
352, 34syl5eq 2655 . 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 1474  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  cin 3538  {csn 4124  cmpt 4637  cfv 5790  (class class class)co 6527  Atomscatm 33371  PSubSpcpsubsp 33603  +𝑃cpadd 33902  𝑃cpolN 34009  WAtomscwpointsN 34093  PAutcpautN 34094  DilcdilN 34209  TrnctrnN 34210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-trnN 34214
This theorem is referenced by:  trnsetN  34264
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