Theorem dvnmptdivc 46061

Description: Function-builder for iterated derivative, division rule for constant divisor. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
dvnmptdivc.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvnmptdivc.x	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
dvnmptdivc.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvnmptdivc.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)
dvnmptdivc.dvn	⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵))
dvnmptdivc.c	⊢ (𝜑 → 𝐶 ∈ ℂ)
dvnmptdivc.cne0	⊢ (𝜑 → 𝐶 ≠ 0)
dvnmptdivc.8	⊢ (𝜑 → 𝑀 ∈ ℕ0)

Assertion

Ref	Expression
dvnmptdivc	⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))

Distinct variable groups:   𝐴,𝑛   𝑥,𝐶   𝑛,𝑀,𝑥   𝑆,𝑛,𝑥   𝑛,𝑋,𝑥   𝜑,𝑛,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑛)   𝐶(𝑛)

Proof of Theorem dvnmptdivc

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → 𝑛 ∈ (0...𝑀))
2		simpl 482	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → 𝜑)
3		fveq2 6828	. . . . 5 ⊢ (𝑘 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0))
4		csbeq1 3849	. . . . . . 7 ⊢ (𝑘 = 0 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋0 / 𝑛⦌𝐵)
5	4	oveq1d 7367	. . . . . 6 ⊢ (𝑘 = 0 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶))
6	5	mpteq2dv 5187	. . . . 5 ⊢ (𝑘 = 0 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
7	3, 6	eqeq12d 2749	. . . 4 ⊢ (𝑘 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶))))
8	7	imbi2d 340	. . 3 ⊢ (𝑘 = 0 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))))
9		fveq2 6828	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))
10		csbeq1 3849	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋𝑗 / 𝑛⦌𝐵)
11	10	oveq1d 7367	. . . . . 6 ⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))
12	11	mpteq2dv 5187	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))
13	9, 12	eqeq12d 2749	. . . 4 ⊢ (𝑘 = 𝑗 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))))
14	13	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑗 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))))
15		fveq2 6828	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
16		csbeq1 3849	. . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → ⦋𝑘 / 𝑛⦌𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵)
17	16	oveq1d 7367	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))
18	17	mpteq2dv 5187	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
19	15, 18	eqeq12d 2749	. . . 4 ⊢ (𝑘 = (𝑗 + 1) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))))
20	19	imbi2d 340	. . 3 ⊢ (𝑘 = (𝑗 + 1) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))))
21		fveq2 6828	. . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛))
22		csbeq1a 3860	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
23	22	equcoms 2021	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
24	23	eqcomd 2739	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ⦋𝑘 / 𝑛⦌𝐵 = 𝐵)
25	24	oveq1d 7367	. . . . . 6 ⊢ (𝑘 = 𝑛 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (𝐵 / 𝐶))
26	25	mpteq2dv 5187	. . . . 5 ⊢ (𝑘 = 𝑛 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))
27	21, 26	eqeq12d 2749	. . . 4 ⊢ (𝑘 = 𝑛 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))))
28	27	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑛 → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶))) ↔ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))))
29		dvnmptdivc.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
30		recnprss 25833	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
31	29, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
32		cnex 11094	. . . . . . . 8 ⊢ ℂ ∈ V
33	32	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
34		dvnmptdivc.a	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
35		dvnmptdivc.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℂ)
36	35	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
37		dvnmptdivc.cne0	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≠ 0)
38	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ≠ 0)
39	34, 36, 38	divcld 11904	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) ∈ ℂ)
40	39	fmpttd 7054	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ)
41		dvnmptdivc.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
42		elpm2r 8775	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
43	33, 29, 40, 41, 42	syl22anc 838	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
44		dvn0 25854	. . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))
45	31, 43, 44	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))
46		id 22	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝜑)
47		dvnmptdivc.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
48		nn0uz 12776	. . . . . . . . . . . . . 14 ⊢ ℕ0 = (ℤ≥‘0)
49	47, 48	eleqtrdi 2843	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
50		eluzfz1 13433	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑀))
51	49, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ (0...𝑀))
52		nfv 1915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 0 ∈ (0...𝑀))
53		nfcv 2895	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)
54		nfcv 2895	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛𝑋
55		nfcsb1v 3870	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋0 / 𝑛⦌𝐵
56	54, 55	nfmpt 5191	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
57	53, 56	nfeq 2909	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
58	52, 57	nfim 1897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
59		c0ex 11113	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
60		eleq1 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...𝑀) ↔ 0 ∈ (0...𝑀)))
61	60	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 0 ∈ (0...𝑀))))
62		fveq2 6828	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0))
63		csbeq1a 3860	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → 𝐵 = ⦋0 / 𝑛⦌𝐵)
64	63	mpteq2dv 5187	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
65	62, 64	eqeq12d 2749	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)))
66	61, 65	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑛 = 0 → (((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))))
67		dvnmptdivc.dvn	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵))
68	58, 59, 66, 67	vtoclf 3518	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
69	46, 51, 68	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
70	69	fveq1d 6830	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥))
71	70	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥))
72		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
73		simpl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝜑)
74	51	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ (0...𝑀))
75		0re 11121	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
76		nfcv 2895	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛0
77		nfv 1915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀))
78		nfcv 2895	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛ℂ
79	55, 78	nfel 2910	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋0 / 𝑛⦌𝐵 ∈ ℂ
80	77, 79	nfim 1897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)
81	60	3anbi3d 1444	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀))))
82	63	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → (𝐵 ∈ ℂ ↔ ⦋0 / 𝑛⦌𝐵 ∈ ℂ))
83	81, 82	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑛 = 0 → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)))
84		dvnmptdivc.b	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)
85	76, 80, 83, 84	vtoclgf 3522	. . . . . . . . . . . 12 ⊢ (0 ∈ ℝ → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ))
86	75, 85	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)
87	73, 72, 74, 86	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)
88		eqid 2733	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
89	88	fvmpt2 6946	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ∧ ⦋0 / 𝑛⦌𝐵 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥) = ⦋0 / 𝑛⦌𝐵)
90	72, 87, 89	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥) = ⦋0 / 𝑛⦌𝐵)
91	71, 90	eqtr2d 2769	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 = (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥))
92	34	fmpttd 7054	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
93		elpm2r 8775	. . . . . . . . . . . 12 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
94	33, 29, 92, 41, 93	syl22anc 838	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
95		dvn0 25854	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ 𝐴))
96	31, 94, 95	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ 𝐴))
97	96	fveq1d 6830	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥))
98	97	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥))
99		eqid 2733	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
100	99	fvmpt2 6946	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
101	72, 34, 100	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
102	91, 98, 101	3eqtrrd 2773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 = ⦋0 / 𝑛⦌𝐵)
103	102	oveq1d 7367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶))
104	103	mpteq2dva 5186	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
105	45, 104	eqtrd 2768	. . . 4 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
106	105	a1i 11	. . 3 ⊢ (𝑀 ∈ (ℤ≥‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶))))
107		simp3 1138	. . . . 5 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
108		simp1 1136	. . . . 5 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝑗 ∈ (0..^𝑀))
109		simpr 484	. . . . . . 7 ⊢ (((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
110		simpl 482	. . . . . . 7 ⊢ (((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))))
111	109, 110	mpd 15	. . . . . 6 ⊢ (((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))
112	111	3adant1 1130	. . . . 5 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))
113	31	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑆 ⊆ ℂ)
114	43	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
115		elfzofz 13577	. . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
116		elfznn0 13522	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
117	116	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
118	115, 117	sylanl2 681	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
119		dvnp1 25855	. . . . . . 7 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
120	113, 114, 118, 119	syl3anc 1373	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
121		oveq2 7360	. . . . . . 7 ⊢ (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))))
122	121	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))))
123	31	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
124	43	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
125		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
126	125, 116	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ0)
127	115, 126	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ ℕ0)
128	123, 124, 127, 119	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
129	128	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
130	29	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
131		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑗 ∈ (0...𝑀))
132	46	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝜑)
133		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
134	132, 133, 131	3jca 1128	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)))
135		nfcv 2895	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝑗
136		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀))
137	135	nfcsb1 3869	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵
138	137, 78	nfel 2910	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ
139	136, 138	nfim 1897	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)
140		eleq1 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑛 ∈ (0...𝑀) ↔ 𝑗 ∈ (0...𝑀)))
141	140	3anbi3d 1444	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀))))
142		csbeq1a 3860	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑛⦌𝐵)
143	142	eleq1d 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ))
144	141, 143	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)))
145	135, 139, 144, 84	vtoclgf 3522	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ))
146	131, 134, 145	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)
147	115, 146	sylanl2 681	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)
148		fzofzp1 13666	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
149	148	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝑗 + 1) ∈ (0...𝑀))
150	115, 132	sylanl2 681	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝜑)
151		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
152	150, 151, 149	3jca 1128	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)))
153		nfcv 2895	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝑗 + 1)
154		nfv 1915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))
155	153	nfcsb1 3869	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵
156	155, 78	nfel 2910	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ
157	154, 156	nfim 1897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)
158		eleq1 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → (𝑛 ∈ (0...𝑀) ↔ (𝑗 + 1) ∈ (0...𝑀)))
159	158	3anbi3d 1444	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))))
160		csbeq1a 3860	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → 𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵)
161	160	eleq1d 2818	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → (𝐵 ∈ ℂ ↔ ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ))
162	159, 161	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑗 + 1) → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)))
163	153, 157, 162, 84	vtoclgf 3522	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ))
164	149, 152, 163	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)
165		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝜑)
166	115	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
167		nfv 1915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑗 ∈ (0...𝑀))
168		nfcv 2895	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)
169	54, 137	nfmpt 5191	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)
170	168, 169	nfeq 2909	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)
171	167, 170	nfim 1897	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
172	140	anbi2d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑀))))
173		fveq2 6828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))
174	142	mpteq2dv 5187	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
175	173, 174	eqeq12d 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)))
176	172, 175	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))))
177	171, 176, 67	chvarfv 2245	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
178	165, 166, 177	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
179	178	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))
180	179	oveq2d 7368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)))
181	165, 94	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
182		dvnp1 25855	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆) ∧ 𝑗 ∈ ℕ0) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)))
183	123, 181, 127, 182	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)))
184	183	eqcomd 2739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)))
185	148	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
186	165, 185	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)))
187		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))
188		nfcv 2895	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1))
189	54, 155	nfmpt 5191	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)
190	188, 189	nfeq 2909	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)
191	187, 190	nfim 1897	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
192	158	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))))
193		fveq2 6828	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)))
194	160	mpteq2dv 5187	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
195	193, 194	eqeq12d 2749	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)))
196	192, 195	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → (((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))))
197	153, 191, 196, 67	vtoclgf 3522	. . . . . . . . . . . . 13 ⊢ ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)))
198	185, 186, 197	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
199	180, 184, 198	3eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
200	35	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝐶 ∈ ℂ)
201	37	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝐶 ≠ 0)
202	130, 147, 164, 199, 200, 201	dvmptdivc 25897	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
203	202	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
204	129, 122, 203	3eqtrd 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
205	204	eqcomd 2739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
206	205, 120, 122	3eqtrrd 2773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
207	120, 122, 206	3eqtrd 2772	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
208	107, 108, 112, 207	syl21anc 837	. . . 4 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
209	208	3exp 1119	. . 3 ⊢ (𝑗 ∈ (0..^𝑀) → ((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))))
210	8, 14, 20, 28, 106, 209	fzind2 13690	. 2 ⊢ (𝑛 ∈ (0...𝑀) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))))
211	1, 2, 210	sylc 65	1 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  Vcvv 3437  ⦋csb 3846   ⊆ wss 3898  {cpr 4577   ↦ cmpt 5174  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ↑_pm cpm 8757  ℂcc 11011  ℝcr 11012  0cc0 11013  1c1 11014   + caddc 11016   / cdiv 11781  ℕ₀cn0 12388  ℤ_≥cuz 12738  ...cfz 13409  ..^cfzo 13556   D cdv 25792   D^𝑛 cdvn 25793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091  ax-addf 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-pm 8759  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-fi 9302  df-sup 9333  df-inf 9334  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-q 12849  df-rp 12893  df-xneg 13013  df-xadd 13014  df-xmul 13015  df-icc 13254  df-fz 13410  df-fzo 13557  df-seq 13911  df-exp 13971  df-hash 14240  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-starv 17178  df-sca 17179  df-vsca 17180  df-ip 17181  df-tset 17182  df-ple 17183  df-ds 17185  df-unif 17186  df-hom 17187  df-cco 17188  df-rest 17328  df-topn 17329  df-0g 17347  df-gsum 17348  df-topgen 17349  df-pt 17350  df-prds 17353  df-xrs 17408  df-qtop 17413  df-imas 17414  df-xps 17416  df-mre 17490  df-mrc 17491  df-acs 17493  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-submnd 18694  df-mulg 18983  df-cntz 19231  df-cmn 19696  df-psmet 21285  df-xmet 21286  df-met 21287  df-bl 21288  df-mopn 21289  df-fbas 21290  df-fg 21291  df-cnfld 21294  df-top 22810  df-topon 22827  df-topsp 22849  df-bases 22862  df-cld 22935  df-ntr 22936  df-cls 22937  df-nei 23014  df-lp 23052  df-perf 23053  df-cn 23143  df-cnp 23144  df-haus 23231  df-tx 23478  df-hmeo 23671  df-fil 23762  df-fm 23854  df-flim 23855  df-flf 23856  df-xms 24236  df-ms 24237  df-tms 24238  df-cncf 24799  df-limc 25795  df-dv 25796  df-dvn 25797

This theorem is referenced by: (None)