Theorem psrval 19869

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	⊢ + = (+g‘𝑅)
psrval.m	⊢ · = (.r‘𝑅)
psrval.o	⊢ 𝑂 = (TopOpen‘𝑅)
psrval.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
psrval.b	⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 𝐷))
psrval.p	⊢ ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵))
psrval.t	⊢ × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))
psrval.v	⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))
psrval.j	⊢ (𝜑 → 𝐽 = (∏t‘(𝐷 × {𝑂})))
psrval.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
psrval.r	⊢ (𝜑 → 𝑅 ∈ 𝑋)

Assertion

Ref	Expression
psrval	⊢ (𝜑 → 𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))

Distinct variable groups:   𝑦,ℎ   𝑓,𝑔,𝑘,𝑥,𝜑   𝐵,𝑓,𝑔,𝑘,𝑥   𝑓,ℎ,𝐼,𝑔,𝑘,𝑥   𝑅,𝑓,𝑔,𝑘,𝑥   𝑦,𝑓,𝐷,𝑔,𝑘,𝑥

Allowed substitution hints:   𝜑(𝑦,ℎ)   𝐵(𝑦,ℎ)   𝐷(ℎ)   + (𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   ✚ (𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   𝑅(𝑦,ℎ)   𝑆(𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   ∙ (𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   · (𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   × (𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   𝐼(𝑦)   𝐽(𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   𝐾(𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   𝑂(𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   𝑊(𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)   𝑋(𝑥,𝑦,𝑓,𝑔,ℎ,𝑘)

Proof of Theorem psrval

Dummy variables 𝑖 𝑟 𝑏 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrval.s	. 2 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		df-psr 19863	. . . 4 ⊢ mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋{ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
3	2	a1i 11	. . 3 ⊢ (𝜑 → mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋{ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩})))
4		simprl 759	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → 𝑖 = 𝐼)
5	4	oveq2d 6991	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → (ℕ0 ↑𝑚 𝑖) = (ℕ0 ↑𝑚 𝐼))
6		rabeq 3401	. . . . . . 7 ⊢ ((ℕ0 ↑𝑚 𝑖) = (ℕ0 ↑𝑚 𝐼) → {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
7	5, 6	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
8		psrval.d	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
9	7, 8	syl6eqr 2827	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
10	9	csbeq1d 3788	. . . 4 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → ⦋{ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ⦋𝐷 / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
11		ovex 7007	. . . . . . 7 ⊢ (ℕ0 ↑𝑚 𝑖) ∈ V
12	11	rabex 5088	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
13	9, 12	syl6eqelr 2870	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → 𝐷 ∈ V)
14		simplrr 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑟 = 𝑅)
15	14	fveq2d 6501	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = (Base‘𝑅))
16		psrval.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝑅)
17	15, 16	syl6eqr 2827	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → (Base‘𝑟) = 𝐾)
18		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
19	17, 18	oveq12d 6993	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑𝑚 𝑑) = (𝐾 ↑𝑚 𝐷))
20		psrval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝐾 ↑𝑚 𝐷))
21	20	ad2antrr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 = (𝐾 ↑𝑚 𝐷))
22	19, 21	eqtr4d 2812	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ((Base‘𝑟) ↑𝑚 𝑑) = 𝐵)
23	22	csbeq1d 3788	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ⦋𝐵 / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}))
24		ovex 7007	. . . . . . . 8 ⊢ ((Base‘𝑟) ↑𝑚 𝑑) ∈ V
25	22, 24	syl6eqelr 2870	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → 𝐵 ∈ V)
26		simpr 477	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
27	26	opeq2d 4681	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
28	14	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑟 = 𝑅)
29	28	fveq2d 6501	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g‘𝑟) = (+g‘𝑅))
30		psrval.a	. . . . . . . . . . . . . 14 ⊢ + = (+g‘𝑅)
31	29, 30	syl6eqr 2827	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (+g‘𝑟) = + )
32		ofeq 7228	. . . . . . . . . . . . 13 ⊢ ((+g‘𝑟) = + → ∘𝑓 (+g‘𝑟) = ∘𝑓 + )
33	31, 32	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘𝑓 (+g‘𝑟) = ∘𝑓 + )
34	26, 26	xpeq12d 5435	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
35	33, 34	reseq12d 5694	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))
36		psrval.p	. . . . . . . . . . 11 ⊢ ✚ = ( ∘𝑓 + ↾ (𝐵 × 𝐵))
37	35, 36	syl6eqr 2827	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏)) = ✚ )
38	37	opeq2d 4681	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩ = ⟨(+g‘ndx), ✚ ⟩)
39	18	adantr 473	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑑 = 𝐷)
40		rabeq 3401	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝐷 → {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘})
41	39, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘})
42	28	fveq2d 6501	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r‘𝑟) = (.r‘𝑅))
43		psrval.m	. . . . . . . . . . . . . . . . 17 ⊢ · = (.r‘𝑅)
44	42, 43	syl6eqr 2827	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (.r‘𝑟) = · )
45	44	oveqd 6992	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥))) = ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥))))
46	41, 45	mpteq12dv 5009	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))) = (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥)))))
47	28, 46	oveq12d 6993	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥))))) = (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥))))))
48	39, 47	mpteq12dv 5009	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))) = (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))
49	26, 26, 48	mpoeq123dv 7046	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥))))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥))))))))
50		psrval.t	. . . . . . . . . . 11 ⊢ × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))
51	49, 50	syl6eqr 2827	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥))))))) = × )
52	51	opeq2d 4681	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩ = ⟨(.r‘ndx), × ⟩)
53	27, 38, 52	tpeq123d 4555	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩})
54	28	opeq2d 4681	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(Scalar‘ndx), 𝑟⟩ = ⟨(Scalar‘ndx), 𝑅⟩)
55	17	adantr 473	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (Base‘𝑟) = 𝐾)
56		ofeq 7228	. . . . . . . . . . . . . 14 ⊢ ((.r‘𝑟) = · → ∘𝑓 (.r‘𝑟) = ∘𝑓 · )
57	44, 56	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ∘𝑓 (.r‘𝑟) = ∘𝑓 · )
58	39	xpeq1d 5433	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {𝑥}) = (𝐷 × {𝑥}))
59		eqidd 2774	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
60	57, 58, 59	oveq123d 6996	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓) = ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))
61	55, 26, 60	mpoeq123dv 7046	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓)) = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)))
62		psrval.v	. . . . . . . . . . 11 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))
63	61, 62	syl6eqr 2827	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓)) = ∙ )
64	63	opeq2d 4681	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩ = ⟨( ·𝑠 ‘ndx), ∙ ⟩)
65	28	fveq2d 6501	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = (TopOpen‘𝑅))
66		psrval.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (TopOpen‘𝑅)
67	65, 66	syl6eqr 2827	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (TopOpen‘𝑟) = 𝑂)
68	67	sneqd 4448	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {(TopOpen‘𝑟)} = {𝑂})
69	39, 68	xpeq12d 5435	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (𝑑 × {(TopOpen‘𝑟)}) = (𝐷 × {𝑂}))
70	69	fveq2d 6501	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = (∏t‘(𝐷 × {𝑂})))
71		psrval.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 = (∏t‘(𝐷 × {𝑂})))
72	71	ad3antrrr 718	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → 𝐽 = (∏t‘(𝐷 × {𝑂})))
73	70, 72	eqtr4d 2812	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → (∏t‘(𝑑 × {(TopOpen‘𝑟)})) = 𝐽)
74	73	opeq2d 4681	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
75	54, 64, 74	tpeq123d 4555	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩} = {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76	53, 75	uneq12d 4024	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) ∧ 𝑏 = 𝐵) → ({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
77	25, 76	csbied 3810	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ⦋𝐵 / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
78	23, 77	eqtrd 2809	. . . . 5 ⊢ (((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) ∧ 𝑑 = 𝐷) → ⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
79	13, 78	csbied 3810	. . . 4 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → ⦋𝐷 / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑑 × {(TopOpen‘𝑟)}))⟩}) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
80	10, 79	eqtrd 2809	. . 3 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → ⦋{ℎ ∈ (ℕ0 ↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘𝑓 (+g‘𝑟) ↾ (𝑏 × 𝑏))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑟⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓 (.r‘𝑟)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 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
82	81	elexd 3430	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
83		psrval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑋)
84	83	elexd 3430	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
85		tpex 7286	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∈ V
86		tpex 7286	. . . . 5 ⊢ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
87	85, 86	unex 7285	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) ∈ V
88	87	a1i 11	. . 3 ⊢ (𝜑 → ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}) ∈ V)
89	3, 80, 82, 84, 88	ovmpod 7117	. 2 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
90	1, 89	syl5eq 2821	1 ⊢ (𝜑 → 𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), ∙ ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {crab 3087  Vcvv 3410  ⦋csb 3781   ∪ cun 3822  {csn 4436  {ctp 4440  ⟨cop 4442   class class class wbr 4926   ↦ cmpt 5005   × cxp 5402  ^◡ccnv 5403   ↾ cres 5406   “ cima 5407  ‘cfv 6186  (class class class)co 6975   ∈ cmpo 6977   ∘_𝑓 cof 7224   ∘_𝑟 cofr 7225   ↑_𝑚 cmap 8205  Fincfn 8305   ≤ cle 10474   − cmin 10669  ℕcn 11438  ℕ₀cn0 11706  ndxcnx 16335  Basecbs 16338  +_gcplusg 16420  ._rcmulr 16421  Scalarcsca 16423   ·_𝑠 cvsca 16424  TopSetcts 16426  TopOpenctopn 16550  ∏_tcpt 16567   Σ_g cgsu 16569   mPwSer cmps 19858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183  ax-un 7278

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-res 5416  df-iota 6150  df-fun 6188  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-of 7226  df-psr 19863

This theorem is referenced by:  psrbas  19885  psrplusg  19888  psrmulr  19891  psrsca  19896  psrvscafval  19897