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Theorem psrval 19577
 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 19571 . . . 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 754 . . . . . . . 8 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝑖 = 𝐼)
54oveq2d 6812 . . . . . . 7 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → (ℕ0𝑚 𝑖) = (ℕ0𝑚 𝐼))
6 rabeq 3342 . . . . . . 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 2823 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
109csbeq1d 3689 . . . 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 6827 . . . . . . 7 (ℕ0𝑚 𝑖) ∈ V
1211rabex 4947 . . . . . 6 { ∈ (ℕ0𝑚 𝑖) ∣ ( “ ℕ) ∈ Fin} ∈ V
139, 12syl6eqelr 2859 . . . . 5 ((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) → 𝐷 ∈ V)
14 simplrr 763 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑟 = 𝑅)
1514fveq2d 6337 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = (Base‘𝑅))
16 psrval.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
1715, 16syl6eqr 2823 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = 𝐾)
18 simpr 471 . . . . . . . . 9 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
1917, 18oveq12d 6814 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑𝑚 𝑑) = (𝐾𝑚 𝐷))
20 psrval.b . . . . . . . . 9 (𝜑𝐵 = (𝐾𝑚 𝐷))
2120ad2antrr 705 . . . . . . . 8 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 = (𝐾𝑚 𝐷))
2219, 21eqtr4d 2808 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑𝑚 𝑑) = 𝐵)
2322csbeq1d 3689 . . . . . 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 6827 . . . . . . . 8 ((Base‘𝑟) ↑𝑚 𝑑) ∈ V
2522, 24syl6eqelr 2859 . . . . . . 7 (((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 ∈ V)
26 simpr 471 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
2726opeq2d 4547 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
2814adantr 466 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑟 = 𝑅)
2928fveq2d 6337 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g𝑟) = (+g𝑅))
30 psrval.a . . . . . . . . . . . . . 14 + = (+g𝑅)
3129, 30syl6eqr 2823 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g𝑟) = + )
32 ofeq 7050 . . . . . . . . . . . . 13 ((+g𝑟) = + → ∘𝑓 (+g𝑟) = ∘𝑓 + )
3331, 32syl 17 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘𝑓 (+g𝑟) = ∘𝑓 + )
3426, 26xpeq12d 5280 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
3533, 34reseq12d 5534 . . . . . . . . . . 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 2823 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘𝑓 (+g𝑟) ↾ (𝑏 × 𝑏)) = )
3837opeq2d 4547 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(+g‘ndx), ( ∘𝑓 (+g𝑟) ↾ (𝑏 × 𝑏))⟩ = ⟨(+g‘ndx), ⟩)
3918adantr 466 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
40 rabeq 3342 . . . . . . . . . . . . . . . 16 (𝑑 = 𝐷 → {𝑦𝑑𝑦𝑟𝑘} = {𝑦𝐷𝑦𝑟𝑘})
4139, 40syl 17 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {𝑦𝑑𝑦𝑟𝑘} = {𝑦𝐷𝑦𝑟𝑘})
4228fveq2d 6337 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r𝑟) = (.r𝑅))
43 psrval.m . . . . . . . . . . . . . . . . 17 · = (.r𝑅)
4442, 43syl6eqr 2823 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r𝑟) = · )
4544oveqd 6813 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘𝑓𝑥))) = ((𝑓𝑥) · (𝑔‘(𝑘𝑓𝑥))))
4641, 45mpteq12dv 4868 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ {𝑦𝑑𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘𝑓𝑥)))) = (𝑥 ∈ {𝑦𝐷𝑦𝑟𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘𝑓𝑥)))))
4728, 46oveq12d 6814 . . . . . . . . . . . . 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 4868 . . . . . . . . . . . 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, 48mpt2eq123dv 6868 . . . . . . . . . . 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 2823 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑘𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦𝑑𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑟)(𝑔‘(𝑘𝑓𝑥))))))) = × )
5251opeq2d 4547 . . . . . . . . 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 4420 . . . . . . . 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 4547 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Scalar‘ndx), 𝑟⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
5517adantr 466 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Base‘𝑟) = 𝐾)
56 ofeq 7050 . . . . . . . . . . . . . 14 ((.r𝑟) = · → ∘𝑓 (.r𝑟) = ∘𝑓 · )
5744, 56syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘𝑓 (.r𝑟) = ∘𝑓 · )
5839xpeq1d 5278 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {𝑥}) = (𝐷 × {𝑥}))
59 eqidd 2772 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
6057, 58, 59oveq123d 6817 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑑 × {𝑥}) ∘𝑓 (.r𝑟)𝑓) = ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))
6155, 26, 60mpt2eq123dv 6868 . . . . . . . . . . 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 2823 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r𝑟)𝑓)) = )
6463opeq2d 4547 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r𝑟)𝑓))⟩ = ⟨( ·𝑠 ‘ndx), ⟩)
6528fveq2d 6337 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = (TopOpen‘𝑅))
66 psrval.o . . . . . . . . . . . . . . 15 𝑂 = (TopOpen‘𝑅)
6765, 66syl6eqr 2823 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = 𝑂)
6867sneqd 4329 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {(TopOpen‘𝑟)} = {𝑂})
6939, 68xpeq12d 5280 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {(TopOpen‘𝑟)}) = (𝐷 × {𝑂}))
7069fveq2d 6337 . . . . . . . . . . 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 709 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝐽 = (∏t‘(𝐷 × {𝑂})))
7370, 72eqtr4d 2808 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = 𝐽)
7473opeq2d 4547 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 = 𝐼𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
7554, 64, 74tpeq123d 4420 . . . . . . . 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 3919 . . . . . . 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 3709 . . . . . 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 2805 . . . . 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 3709 . . . 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 2805 . . 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)
82 elex 3364 . . . 4 (𝐼𝑊𝐼 ∈ V)
8381, 82syl 17 . . 3 (𝜑𝐼 ∈ V)
84 psrval.r . . . 4 (𝜑𝑅𝑋)
85 elex 3364 . . . 4 (𝑅𝑋𝑅 ∈ V)
8684, 85syl 17 . . 3 (𝜑𝑅 ∈ V)
87 tpex 7108 . . . . 5 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
88 tpex 7108 . . . . 5 {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
8987, 88unex 7107 . . . 4 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) ∈ V
9089a1i 11 . . 3 (𝜑 → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) ∈ V)
913, 80, 83, 86, 90ovmpt2d 6939 . 2 (𝜑 → (𝐼 mPwSer 𝑅) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
921, 91syl5eq 2817 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 382   = wceq 1631   ∈ wcel 2145  {crab 3065  Vcvv 3351  ⦋csb 3682   ∪ cun 3721  {csn 4317  {ctp 4321  ⟨cop 4323   class class class wbr 4787   ↦ cmpt 4864   × cxp 5248  ◡ccnv 5249   ↾ cres 5252   “ cima 5253  ‘cfv 6030  (class class class)co 6796   ↦ cmpt2 6798   ∘𝑓 cof 7046   ∘𝑟 cofr 7047   ↑𝑚 cmap 8013  Fincfn 8113   ≤ cle 10281   − cmin 10472  ℕcn 11226  ℕ0cn0 11499  ndxcnx 16061  Basecbs 16064  +gcplusg 16149  .rcmulr 16150  Scalarcsca 16152   ·𝑠 cvsca 16153  TopSetcts 16155  TopOpenctopn 16290  ∏tcpt 16307   Σg cgsu 16309   mPwSer cmps 19566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-res 5262  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-of 7048  df-psr 19571 This theorem is referenced by:  psrbas  19593  psrplusg  19596  psrmulr  19599  psrsca  19604  psrvscafval  19605
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