Theorem dvnmptdivc 46293

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 6842	. . . . 5 ⊢ (𝑘 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0))
4		csbeq1 3854	. . . . . . 7 ⊢ (𝑘 = 0 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋0 / 𝑛⦌𝐵)
5	4	oveq1d 7383	. . . . . 6 ⊢ (𝑘 = 0 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶))
6	5	mpteq2dv 5194	. . . . 5 ⊢ (𝑘 = 0 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
7	3, 6	eqeq12d 2753	. . . 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 6842	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))
10		csbeq1 3854	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋𝑗 / 𝑛⦌𝐵)
11	10	oveq1d 7383	. . . . . 6 ⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))
12	11	mpteq2dv 5194	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))
13	9, 12	eqeq12d 2753	. . . 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 6842	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
16		csbeq1 3854	. . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → ⦋𝑘 / 𝑛⦌𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵)
17	16	oveq1d 7383	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))
18	17	mpteq2dv 5194	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
19	15, 18	eqeq12d 2753	. . . 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 6842	. . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛))
22		csbeq1a 3865	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
23	22	equcoms 2022	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
24	23	eqcomd 2743	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ⦋𝑘 / 𝑛⦌𝐵 = 𝐵)
25	24	oveq1d 7383	. . . . . 6 ⊢ (𝑘 = 𝑛 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (𝐵 / 𝐶))
26	25	mpteq2dv 5194	. . . . 5 ⊢ (𝑘 = 𝑛 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))
27	21, 26	eqeq12d 2753	. . . 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 25873	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
31	29, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
32		cnex 11119	. . . . . . . 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 11929	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) ∈ ℂ)
40	39	fmpttd 7069	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ)
41		dvnmptdivc.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
42		elpm2r 8794	. . . . . . 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 839	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
44		dvn0 25894	. . . . . 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 585	. . . . 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 12801	. . . . . . . . . . . . . 14 ⊢ ℕ0 = (ℤ≥‘0)
49	47, 48	eleqtrdi 2847	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
50		eluzfz1 13459	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑀))
51	49, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ (0...𝑀))
52		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 0 ∈ (0...𝑀))
53		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)
54		nfcv 2899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛𝑋
55		nfcsb1v 3875	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋0 / 𝑛⦌𝐵
56	54, 55	nfmpt 5198	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
57	53, 56	nfeq 2913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
58	52, 57	nfim 1898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
59		c0ex 11138	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
60		eleq1 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...𝑀) ↔ 0 ∈ (0...𝑀)))
61	60	anbi2d 631	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 0 ∈ (0...𝑀))))
62		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0))
63		csbeq1a 3865	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → 𝐵 = ⦋0 / 𝑛⦌𝐵)
64	63	mpteq2dv 5194	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
65	62, 64	eqeq12d 2753	. . . . . . . . . . . . . 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 3523	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
69	46, 51, 68	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
70	69	fveq1d 6844	. . . . . . . . . 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 11146	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
76		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛0
77		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀))
78		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛ℂ
79	55, 78	nfel 2914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋0 / 𝑛⦌𝐵 ∈ ℂ
80	77, 79	nfim 1898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)
81	60	3anbi3d 1445	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀))))
82	63	eleq1d 2822	. . . . . . . . . . . . . 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 3527	. . . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)
88		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
89	88	fvmpt2 6961	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ∧ ⦋0 / 𝑛⦌𝐵 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥) = ⦋0 / 𝑛⦌𝐵)
90	72, 87, 89	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥) = ⦋0 / 𝑛⦌𝐵)
91	71, 90	eqtr2d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 = (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥))
92	34	fmpttd 7069	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
93		elpm2r 8794	. . . . . . . . . . . 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 839	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
95		dvn0 25894	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ 𝐴))
96	31, 94, 95	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ 𝐴))
97	96	fveq1d 6844	. . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
100	99	fvmpt2 6961	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
101	72, 34, 100	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
102	91, 98, 101	3eqtrrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 = ⦋0 / 𝑛⦌𝐵)
103	102	oveq1d 7383	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶))
104	103	mpteq2dva 5193	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
105	45, 104	eqtrd 2772	. . . 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 1139	. . . . 5 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
108		simp1 1137	. . . . 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 1131	. . . . 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 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑆 ⊆ ℂ)
114	43	ad2antrr 727	. . . . . . 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 13603	. . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
116		elfznn0 13548	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
117	116	ad2antlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
118	115, 117	sylanl2 682	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
119		dvnp1 25895	. . . . . . 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 1374	. . . . . 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 7376	. . . . . . 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 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ ℕ0)
128	123, 124, 127, 119	syl3anc 1374	. . . . . . . . . 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 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑗 ∈ (0...𝑀))
132	46	ad2antrr 727	. . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)))
135		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝑗
136		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀))
137	135	nfcsb1 3874	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵
138	137, 78	nfel 2914	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ
139	136, 138	nfim 1898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)
140		eleq1 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑛 ∈ (0...𝑀) ↔ 𝑗 ∈ (0...𝑀)))
141	140	3anbi3d 1445	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀))))
142		csbeq1a 3865	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑛⦌𝐵)
143	142	eleq1d 2822	. . . . . . . . . . . . . . 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 3527	. . . . . . . . . . . . 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 682	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)
148		fzofzp1 13692	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
149	148	ad2antlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝑗 + 1) ∈ (0...𝑀))
150	115, 132	sylanl2 682	. . . . . . . . . . . . 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 1129	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)))
153		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝑗 + 1)
154		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))
155	153	nfcsb1 3874	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵
156	155, 78	nfel 2914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ
157	154, 156	nfim 1898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)
158		eleq1 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → (𝑛 ∈ (0...𝑀) ↔ (𝑗 + 1) ∈ (0...𝑀)))
159	158	3anbi3d 1445	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))))
160		csbeq1a 3865	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → 𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵)
161	160	eleq1d 2822	. . . . . . . . . . . . . 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 3527	. . . . . . . . . . . 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 1916	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑗 ∈ (0...𝑀))
168		nfcv 2899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)
169	54, 137	nfmpt 5198	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)
170	168, 169	nfeq 2913	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)
171	167, 170	nfim 1898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
172	140	anbi2d 631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑀))))
173		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))
174	142	mpteq2dv 5194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
175	173, 174	eqeq12d 2753	. . . . . . . . . . . . . . . . 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 2248	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
178	165, 166, 177	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
179	178	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))
180	179	oveq2d 7384	. . . . . . . . . . . 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 25895	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)))
184	183	eqcomd 2743	. . . . . . . . . . . 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 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))
188		nfcv 2899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1))
189	54, 155	nfmpt 5198	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)
190	188, 189	nfeq 2913	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)
191	187, 190	nfim 1898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
192	158	anbi2d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))))
193		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)))
194	160	mpteq2dv 5194	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
195	193, 194	eqeq12d 2753	. . . . . . . . . . . . . . 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 3527	. . . . . . . . . . . . 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 2776	. . . . . . . . . . 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 25937	. . . . . . . . . 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 2776	. . . . . . . 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 2743	. . . . . . 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 2777	. . . . . 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 2776	. . . . 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 838	. . . 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 1120	. . 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 13716	. 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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442  ⦋csb 3851   ⊆ wss 3903  {cpr 4584   ↦ cmpt 5181  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_pm cpm 8776  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   / cdiv 11806  ℕ₀cn0 12413  ℤ_≥cuz 12763  ...cfz 13435  ..^cfzo 13582   D cdv 25832   D^𝑛 cdvn 25833

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-fi 9326  df-sup 9357  df-inf 9358  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-icc 13280  df-fz 13436  df-fzo 13583  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-starv 17204  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-unif 17212  df-hom 17213  df-cco 17214  df-rest 17354  df-topn 17355  df-0g 17373  df-gsum 17374  df-topgen 17375  df-pt 17376  df-prds 17379  df-xrs 17435  df-qtop 17440  df-imas 17441  df-xps 17443  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-mulg 19010  df-cntz 19258  df-cmn 19723  df-psmet 21313  df-xmet 21314  df-met 21315  df-bl 21316  df-mopn 21317  df-fbas 21318  df-fg 21319  df-cnfld 21322  df-top 22850  df-topon 22867  df-topsp 22889  df-bases 22902  df-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054  df-lp 23092  df-perf 23093  df-cn 23183  df-cnp 23184  df-haus 23271  df-tx 23518  df-hmeo 23711  df-fil 23802  df-fm 23894  df-flim 23895  df-flf 23896  df-xms 24276  df-ms 24277  df-tms 24278  df-cncf 24839  df-limc 25835  df-dv 25836  df-dvn 25837

This theorem is referenced by: (None)