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Theorem psrval 20142
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 20136 . . . 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 770 . . . . . . . 8 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝑖 = 𝐼)
54oveq2d 7165 . . . . . . 7 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → (ℕ0m 𝑖) = (ℕ0m 𝐼))
6 rabeq 3469 . . . . . . 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 2877 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
109csbeq1d 3870 . . . 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 7182 . . . . . . 7 (ℕ0m 𝑖) ∈ V
1211rabex 5221 . . . . . 6 { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∈ V
139, 12eqeltrrdi 2925 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝐷 ∈ V)
14 simplrr 777 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑟 = 𝑅)
1514fveq2d 6665 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = (Base‘𝑅))
16 psrval.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
1715, 16syl6eqr 2877 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = 𝐾)
18 simpr 488 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
1917, 18oveq12d 7167 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) = (𝐾m 𝐷))
20 psrval.b . . . . . . . . 9 (𝜑𝐵 = (𝐾m 𝐷))
2120ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 = (𝐾m 𝐷))
2219, 21eqtr4d 2862 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) = 𝐵)
2322csbeq1d 3870 . . . . . 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 7182 . . . . . . . 8 ((Base‘𝑟) ↑m 𝑑) ∈ V
2522, 24eqeltrrdi 2925 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 ∈ V)
26 simpr 488 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2726opeq2d 4796 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
2814adantr 484 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑟 = 𝑅)
2928fveq2d 6665 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g𝑟) = (+g𝑅))
30 psrval.a . . . . . . . . . . . . . 14 + = (+g𝑅)
3129, 30syl6eqr 2877 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g𝑟) = + )
32 ofeq 7405 . . . . . . . . . . . . 13 ((+g𝑟) = + → ∘f (+g𝑟) = ∘f + )
3331, 32syl 17 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘f (+g𝑟) = ∘f + )
3426, 26xpeq12d 5573 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
3533, 34reseq12d 5841 . . . . . . . . . . 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 2877 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏)) = )
3837opeq2d 4796 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩ = ⟨(+g‘ndx), ⟩)
3918adantr 484 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
40 rabeq 3469 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → {𝑦𝑑𝑦r𝑘} = {𝑦𝐷𝑦r𝑘})
4139, 40syl 17 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {𝑦𝑑𝑦r𝑘} = {𝑦𝐷𝑦r𝑘})
4228fveq2d 6665 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r𝑟) = (.r𝑅))
43 psrval.m . . . . . . . . . . . . . . . . 17 · = (.r𝑅)
4442, 43syl6eqr 2877 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r𝑟) = · )
4544oveqd 7166 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))) = ((𝑓𝑥) · (𝑔‘(𝑘f𝑥))))
4641, 45mpteq12dv 5137 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))) = (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))
4728, 46oveq12d 7167 . . . . . . . . . . . . 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 5137 . . . . . . . . . . . 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 7222 . . . . . . . . . . 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 2877 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))))))) = × )
5251opeq2d 4796 . . . . . . . . 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 4669 . . . . . . . 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 4796 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Scalar‘ndx), 𝑟⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
5517adantr 484 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Base‘𝑟) = 𝐾)
56 ofeq 7405 . . . . . . . . . . . . . 14 ((.r𝑟) = · → ∘f (.r𝑟) = ∘f · )
5744, 56syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘f (.r𝑟) = ∘f · )
5839xpeq1d 5571 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {𝑥}) = (𝐷 × {𝑥}))
59 eqidd 2825 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
6057, 58, 59oveq123d 7170 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓) = ((𝐷 × {𝑥}) ∘f · 𝑓))
6155, 26, 60mpoeq123dv 7222 . . . . . . . . . . 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 2877 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓)) = )
6463opeq2d 4796 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩ = ⟨( ·𝑠 ‘ndx), ⟩)
6528fveq2d 6665 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = (TopOpen‘𝑅))
66 psrval.o . . . . . . . . . . . . . . 15 𝑂 = (TopOpen‘𝑅)
6765, 66syl6eqr 2877 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = 𝑂)
6867sneqd 4562 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {(TopOpen‘𝑟)} = {𝑂})
6939, 68xpeq12d 5573 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {(TopOpen‘𝑟)}) = (𝐷 × {𝑂}))
7069fveq2d 6665 . . . . . . . . . . 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 729 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝐽 = (∏t‘(𝐷 × {𝑂})))
7370, 72eqtr4d 2862 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = 𝐽)
7473opeq2d 4796 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
7554, 64, 74tpeq123d 4669 . . . . . . . 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 4126 . . . . . . 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 3902 . . . . . 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 2859 . . . . 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 3902 . . . 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 2859 . . 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 3500 . . 3 (𝜑𝐼 ∈ V)
83 psrval.r . . . 4 (𝜑𝑅𝑋)
8483elexd 3500 . . 3 (𝜑𝑅 ∈ V)
85 tpex 7464 . . . . 5 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
86 tpex 7464 . . . . 5 {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
8785, 86unex 7463 . . . 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 7295 . 2 (𝜑 → (𝐼 mPwSer 𝑅) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
901, 89syl5eq 2871 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 399   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  csb 3866  cun 3917  {csn 4550  {ctp 4554  cop 4556   class class class wbr 5052  cmpt 5132   × cxp 5540  ccnv 5541  cres 5544  cima 5545  cfv 6343  (class class class)co 7149  cmpo 7151  f cof 7401  r cofr 7402  m cmap 8402  Fincfn 8505  cle 10674  cmin 10868  cn 11634  0cn0 11894  ndxcnx 16480  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  Scalarcsca 16568   ·𝑠 cvsca 16569  TopSetcts 16571  TopOpenctopn 16695  tcpt 16712   Σg cgsu 16714   mPwSer cmps 20131
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-psr 20136
This theorem is referenced by:  psrbas  20158  psrplusg  20161  psrmulr  20164  psrsca  20169  psrvscafval  20170
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