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Theorem psrval 20070
Description: Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
psrval.s 𝑆 = (𝐼 mPwSer 𝑅)
psrval.k 𝐾 = (Base‘𝑅)
psrval.a + = (+g𝑅)
psrval.m · = (.r𝑅)
psrval.o 𝑂 = (TopOpen‘𝑅)
psrval.d 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
psrval.b (𝜑𝐵 = (𝐾m 𝐷))
psrval.p = ( ∘f + ↾ (𝐵 × 𝐵))
psrval.t × = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))))
psrval.v = (𝑥𝐾, 𝑓𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))
psrval.j (𝜑𝐽 = (∏t‘(𝐷 × {𝑂})))
psrval.i (𝜑𝐼𝑊)
psrval.r (𝜑𝑅𝑋)
Assertion
Ref Expression
psrval (𝜑𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
Distinct variable groups:   𝑦,   𝑓,𝑔,𝑘,𝑥,𝜑   𝐵,𝑓,𝑔,𝑘,𝑥   𝑓,,𝐼,𝑔,𝑘,𝑥   𝑅,𝑓,𝑔,𝑘,𝑥   𝑦,𝑓,𝐷,𝑔,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑦,)   𝐵(𝑦,)   𝐷()   + (𝑥,𝑦,𝑓,𝑔,,𝑘)   (𝑥,𝑦,𝑓,𝑔,,𝑘)   𝑅(𝑦,)   𝑆(𝑥,𝑦,𝑓,𝑔,,𝑘)   (𝑥,𝑦,𝑓,𝑔,,𝑘)   · (𝑥,𝑦,𝑓,𝑔,,𝑘)   × (𝑥,𝑦,𝑓,𝑔,,𝑘)   𝐼(𝑦)   𝐽(𝑥,𝑦,𝑓,𝑔,,𝑘)   𝐾(𝑥,𝑦,𝑓,𝑔,,𝑘)   𝑂(𝑥,𝑦,𝑓,𝑔,,𝑘)   𝑊(𝑥,𝑦,𝑓,𝑔,,𝑘)   𝑋(𝑥,𝑦,𝑓,𝑔,,𝑘)

Proof of Theorem psrval
Dummy variables 𝑖 𝑟 𝑏 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrval.s . 2 𝑆 = (𝐼 mPwSer 𝑅)
2 df-psr 20064 . . . 4 mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
32a1i 11 . . 3 (𝜑 → mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩})))
4 simprl 767 . . . . . . . 8 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝑖 = 𝐼)
54oveq2d 7161 . . . . . . 7 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → (ℕ0m 𝑖) = (ℕ0m 𝐼))
6 rabeq 3481 . . . . . . 7 ((ℕ0m 𝑖) = (ℕ0m 𝐼) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
75, 6syl 17 . . . . . 6 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
8 psrval.d . . . . . 6 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
97, 8syl6eqr 2871 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
109csbeq1d 3884 . . . 4 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = 𝐷 / 𝑑((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
11 ovex 7178 . . . . . . 7 (ℕ0m 𝑖) ∈ V
1211rabex 5226 . . . . . 6 { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∈ V
139, 12syl6eqelr 2919 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝐷 ∈ V)
14 simplrr 774 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑟 = 𝑅)
1514fveq2d 6667 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = (Base‘𝑅))
16 psrval.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
1715, 16syl6eqr 2871 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = 𝐾)
18 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
1917, 18oveq12d 7163 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) = (𝐾m 𝐷))
20 psrval.b . . . . . . . . 9 (𝜑𝐵 = (𝐾m 𝐷))
2120ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 = (𝐾m 𝐷))
2219, 21eqtr4d 2856 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) = 𝐵)
2322csbeq1d 3884 . . . . . 6 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = 𝐵 / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
24 ovex 7178 . . . . . . . 8 ((Base‘𝑟) ↑m 𝑑) ∈ V
2522, 24syl6eqelr 2919 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 ∈ V)
26 simpr 485 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2726opeq2d 4802 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
2814adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑟 = 𝑅)
2928fveq2d 6667 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g𝑟) = (+g𝑅))
30 psrval.a . . . . . . . . . . . . . 14 + = (+g𝑅)
3129, 30syl6eqr 2871 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g𝑟) = + )
32 ofeq 7400 . . . . . . . . . . . . 13 ((+g𝑟) = + → ∘f (+g𝑟) = ∘f + )
3331, 32syl 17 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘f (+g𝑟) = ∘f + )
3426, 26xpeq12d 5579 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
3533, 34reseq12d 5847 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏)) = ( ∘f + ↾ (𝐵 × 𝐵)))
36 psrval.p . . . . . . . . . . 11 = ( ∘f + ↾ (𝐵 × 𝐵))
3735, 36syl6eqr 2871 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏)) = )
3837opeq2d 4802 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩ = ⟨(+g‘ndx), ⟩)
3918adantr 481 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
40 rabeq 3481 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → {𝑦𝑑𝑦r𝑘} = {𝑦𝐷𝑦r𝑘})
4139, 40syl 17 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {𝑦𝑑𝑦r𝑘} = {𝑦𝐷𝑦r𝑘})
4228fveq2d 6667 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r𝑟) = (.r𝑅))
43 psrval.m . . . . . . . . . . . . . . . . 17 · = (.r𝑅)
4442, 43syl6eqr 2871 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r𝑟) = · )
4544oveqd 7162 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))) = ((𝑓𝑥) · (𝑔‘(𝑘f𝑥))))
4641, 45mpteq12dv 5142 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))) = (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))
4728, 46oveq12d 7163 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))))) = (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥))))))
4839, 47mpteq12dv 5142 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))) = (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))))
4926, 26, 48mpoeq123dv 7218 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥))))))))
50 psrval.t . . . . . . . . . . 11 × = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))))
5149, 50syl6eqr 2871 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))))))) = × )
5251opeq2d 4802 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩ = ⟨(.r‘ndx), × ⟩)
5327, 38, 52tpeq123d 4676 . . . . . . . 8 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩})
5428opeq2d 4802 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Scalar‘ndx), 𝑟⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
5517adantr 481 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Base‘𝑟) = 𝐾)
56 ofeq 7400 . . . . . . . . . . . . . 14 ((.r𝑟) = · → ∘f (.r𝑟) = ∘f · )
5744, 56syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘f (.r𝑟) = ∘f · )
5839xpeq1d 5577 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {𝑥}) = (𝐷 × {𝑥}))
59 eqidd 2819 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
6057, 58, 59oveq123d 7166 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓) = ((𝐷 × {𝑥}) ∘f · 𝑓))
6155, 26, 60mpoeq123dv 7218 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓)) = (𝑥𝐾, 𝑓𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)))
62 psrval.v . . . . . . . . . . 11 = (𝑥𝐾, 𝑓𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))
6361, 62syl6eqr 2871 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓)) = )
6463opeq2d 4802 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩ = ⟨( ·𝑠 ‘ndx), ⟩)
6528fveq2d 6667 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = (TopOpen‘𝑅))
66 psrval.o . . . . . . . . . . . . . . 15 𝑂 = (TopOpen‘𝑅)
6765, 66syl6eqr 2871 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = 𝑂)
6867sneqd 4569 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {(TopOpen‘𝑟)} = {𝑂})
6939, 68xpeq12d 5579 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {(TopOpen‘𝑟)}) = (𝐷 × {𝑂}))
7069fveq2d 6667 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = (∏t‘(𝐷 × {𝑂})))
71 psrval.j . . . . . . . . . . . 12 (𝜑𝐽 = (∏t‘(𝐷 × {𝑂})))
7271ad3antrrr 726 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝐽 = (∏t‘(𝐷 × {𝑂})))
7370, 72eqtr4d 2856 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = 𝐽)
7473opeq2d 4802 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
7554, 64, 74tpeq123d 4676 . . . . . . . 8 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩} = {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
7653, 75uneq12d 4137 . . . . . . 7 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
7725, 76csbied 3916 . . . . . 6 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
7823, 77eqtrd 2853 . . . . 5 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
7913, 78csbied 3916 . . . 4 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝐷 / 𝑑((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
8010, 79eqtrd 2853 . . 3 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} / 𝑑((Base‘𝑟) ↑m 𝑑) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
81 psrval.i . . . 4 (𝜑𝐼𝑊)
8281elexd 3512 . . 3 (𝜑𝐼 ∈ V)
83 psrval.r . . . 4 (𝜑𝑅𝑋)
8483elexd 3512 . . 3 (𝜑𝑅 ∈ V)
85 tpex 7459 . . . . 5 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
86 tpex 7459 . . . . 5 {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
8785, 86unex 7458 . . . 4 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) ∈ V
8887a1i 11 . . 3 (𝜑 → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) ∈ V)
893, 80, 82, 84, 88ovmpod 7291 . 2 (𝜑 → (𝐼 mPwSer 𝑅) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
901, 89syl5eq 2865 1 (𝜑𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  csb 3880  cun 3931  {csn 4557  {ctp 4561  cop 4563   class class class wbr 5057  cmpt 5137   × cxp 5546  ccnv 5547  cres 5550  cima 5551  cfv 6348  (class class class)co 7145  cmpo 7147  f cof 7396  r cofr 7397  m cmap 8395  Fincfn 8497  cle 10664  cmin 10858  cn 11626  0cn0 11885  ndxcnx 16468  Basecbs 16471  +gcplusg 16553  .rcmulr 16554  Scalarcsca 16556   ·𝑠 cvsca 16557  TopSetcts 16559  TopOpenctopn 16683  tcpt 16700   Σg cgsu 16702   mPwSer cmps 20059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-psr 20064
This theorem is referenced by:  psrbas  20086  psrplusg  20089  psrmulr  20092  psrsca  20097  psrvscafval  20098
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