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Theorem psrval 21317
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 21311 . . . 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 769 . . . . . . . 8 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝑖 = 𝐼)
54oveq2d 7373 . . . . . . 7 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → (ℕ0m 𝑖) = (ℕ0m 𝐼))
6 rabeq 3421 . . . . . . 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, 8eqtr4di 2794 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
109csbeq1d 3859 . . . 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 7390 . . . . . . 7 (ℕ0m 𝑖) ∈ V
1211rabex 5289 . . . . . 6 { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∈ V
139, 12eqeltrrdi 2847 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝐷 ∈ V)
14 simplrr 776 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑟 = 𝑅)
1514fveq2d 6846 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = (Base‘𝑅))
16 psrval.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
1715, 16eqtr4di 2794 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = 𝐾)
18 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
1917, 18oveq12d 7375 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) = (𝐾m 𝐷))
20 psrval.b . . . . . . . . 9 (𝜑𝐵 = (𝐾m 𝐷))
2120ad2antrr 724 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 = (𝐾m 𝐷))
2219, 21eqtr4d 2779 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑m 𝑑) = 𝐵)
2322csbeq1d 3859 . . . . . 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 7390 . . . . . . . 8 ((Base‘𝑟) ↑m 𝑑) ∈ V
2522, 24eqeltrrdi 2847 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 ∈ V)
26 simpr 485 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2726opeq2d 4837 . . . . . . . . 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 6846 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g𝑟) = (+g𝑅))
30 psrval.a . . . . . . . . . . . . . 14 + = (+g𝑅)
3129, 30eqtr4di 2794 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g𝑟) = + )
3231ofeqd 7619 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘f (+g𝑟) = ∘f + )
3326, 26xpeq12d 5664 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
3432, 33reseq12d 5938 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏)) = ( ∘f + ↾ (𝐵 × 𝐵)))
35 psrval.p . . . . . . . . . . 11 = ( ∘f + ↾ (𝐵 × 𝐵))
3634, 35eqtr4di 2794 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏)) = )
3736opeq2d 4837 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(+g‘ndx), ( ∘f (+g𝑟) ↾ (𝑏 × 𝑏))⟩ = ⟨(+g‘ndx), ⟩)
3818adantr 481 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
39 rabeq 3421 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → {𝑦𝑑𝑦r𝑘} = {𝑦𝐷𝑦r𝑘})
4038, 39syl 17 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {𝑦𝑑𝑦r𝑘} = {𝑦𝐷𝑦r𝑘})
4128fveq2d 6846 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r𝑟) = (.r𝑅))
42 psrval.m . . . . . . . . . . . . . . . . 17 · = (.r𝑅)
4341, 42eqtr4di 2794 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r𝑟) = · )
4443oveqd 7374 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))) = ((𝑓𝑥) · (𝑔‘(𝑘f𝑥))))
4540, 44mpteq12dv 5196 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))) = (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))
4628, 45oveq12d 7375 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))))) = (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥))))))
4738, 46mpteq12dv 5196 . . . . . . . . . . . 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)))))))
4826, 26, 47mpoeq123dv 7432 . . . . . . . . . . 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))))))))
49 psrval.t . . . . . . . . . . 11 × = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))))
5048, 49eqtr4di 2794 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥))))))) = × )
5150opeq2d 4837 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(.r‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦r𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘f𝑥)))))))⟩ = ⟨(.r‘ndx), × ⟩)
5227, 37, 51tpeq123d 4709 . . . . . . . 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), × ⟩})
5328opeq2d 4837 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Scalar‘ndx), 𝑟⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
5417adantr 481 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Base‘𝑟) = 𝐾)
5543ofeqd 7619 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘f (.r𝑟) = ∘f · )
5638xpeq1d 5662 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {𝑥}) = (𝐷 × {𝑥}))
57 eqidd 2737 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
5855, 56, 57oveq123d 7378 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓) = ((𝐷 × {𝑥}) ∘f · 𝑓))
5954, 26, 58mpoeq123dv 7432 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓)) = (𝑥𝐾, 𝑓𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)))
60 psrval.v . . . . . . . . . . 11 = (𝑥𝐾, 𝑓𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))
6159, 60eqtr4di 2794 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓)) = )
6261opeq2d 4837 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘f (.r𝑟)𝑓))⟩ = ⟨( ·𝑠 ‘ndx), ⟩)
6328fveq2d 6846 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = (TopOpen‘𝑅))
64 psrval.o . . . . . . . . . . . . . . 15 𝑂 = (TopOpen‘𝑅)
6563, 64eqtr4di 2794 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = 𝑂)
6665sneqd 4598 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {(TopOpen‘𝑟)} = {𝑂})
6738, 66xpeq12d 5664 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {(TopOpen‘𝑟)}) = (𝐷 × {𝑂}))
6867fveq2d 6846 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = (∏t‘(𝐷 × {𝑂})))
69 psrval.j . . . . . . . . . . . 12 (𝜑𝐽 = (∏t‘(𝐷 × {𝑂})))
7069ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝐽 = (∏t‘(𝐷 × {𝑂})))
7168, 70eqtr4d 2779 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = 𝐽)
7271opeq2d 4837 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
7353, 62, 72tpeq123d 4709 . . . . . . . 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⟩})
7452, 73uneq12d 4124 . . . . . . 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⟩}))
7525, 74csbied 3893 . . . . . 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⟩}))
7623, 75eqtrd 2776 . . . . 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⟩}))
7713, 76csbied 3893 . . . 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⟩}))
7810, 77eqtrd 2776 . . 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⟩}))
79 psrval.i . . . 4 (𝜑𝐼𝑊)
8079elexd 3465 . . 3 (𝜑𝐼 ∈ V)
81 psrval.r . . . 4 (𝜑𝑅𝑋)
8281elexd 3465 . . 3 (𝜑𝑅 ∈ V)
83 tpex 7681 . . . . 5 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
84 tpex 7681 . . . . 5 {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
8583, 84unex 7680 . . . 4 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) ∈ V
8685a1i 11 . . 3 (𝜑 → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) ∈ V)
873, 78, 80, 82, 86ovmpod 7507 . 2 (𝜑 → (𝐼 mPwSer 𝑅) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
881, 87eqtrid 2788 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 1541  wcel 2106  {crab 3407  Vcvv 3445  csb 3855  cun 3908  {csn 4586  {ctp 4590  cop 4592   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  cres 5635  cima 5636  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615  r cofr 7616  m cmap 8765  Fincfn 8883  cle 11190  cmin 11385  cn 12153  0cn0 12413  ndxcnx 17065  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  TopSetcts 17139  TopOpenctopn 17303  tcpt 17320   Σg cgsu 17322   mPwSer cmps 21306
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-res 5645  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-psr 21311
This theorem is referenced by:  psrbas  21346  psrplusg  21349  psrmulr  21352  psrsca  21357  psrvscafval  21358
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