Theorem dvnmptdivc 46366

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 6840	. . . . 5 ⊢ (𝑘 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0))
4		csbeq1 3840	. . . . . . 7 ⊢ (𝑘 = 0 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋0 / 𝑛⦌𝐵)
5	4	oveq1d 7382	. . . . . 6 ⊢ (𝑘 = 0 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶))
6	5	mpteq2dv 5179	. . . . 5 ⊢ (𝑘 = 0 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
7	3, 6	eqeq12d 2752	. . . 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 6840	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))
10		csbeq1 3840	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋𝑗 / 𝑛⦌𝐵)
11	10	oveq1d 7382	. . . . . 6 ⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))
12	11	mpteq2dv 5179	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))
13	9, 12	eqeq12d 2752	. . . 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 6840	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
16		csbeq1 3840	. . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → ⦋𝑘 / 𝑛⦌𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵)
17	16	oveq1d 7382	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))
18	17	mpteq2dv 5179	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
19	15, 18	eqeq12d 2752	. . . 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 6840	. . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛))
22		csbeq1a 3851	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
23	22	equcoms 2022	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
24	23	eqcomd 2742	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ⦋𝑘 / 𝑛⦌𝐵 = 𝐵)
25	24	oveq1d 7382	. . . . . 6 ⊢ (𝑘 = 𝑛 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (𝐵 / 𝐶))
26	25	mpteq2dv 5179	. . . . 5 ⊢ (𝑘 = 𝑛 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))
27	21, 26	eqeq12d 2752	. . . 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 25871	. . . . . . 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 11931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) ∈ ℂ)
40	39	fmpttd 7067	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ)
41		dvnmptdivc.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
42		elpm2r 8792	. . . . . . 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 25891	. . . . . 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 12826	. . . . . . . . . . . . . 14 ⊢ ℕ0 = (ℤ≥‘0)
49	47, 48	eleqtrdi 2846	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
50		eluzfz1 13485	. . . . . . . . . . . . 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 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)
54		nfcv 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛𝑋
55		nfcsb1v 3861	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋0 / 𝑛⦌𝐵
56	54, 55	nfmpt 5183	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
57	53, 56	nfeq 2912	. . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0))
63		csbeq1a 3851	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → 𝐵 = ⦋0 / 𝑛⦌𝐵)
64	63	mpteq2dv 5179	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
65	62, 64	eqeq12d 2752	. . . . . . . . . . . . . 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 3509	. . . . . . . . . . . 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 6842	. . . . . . . . . 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 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛0
77		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀))
78		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛ℂ
79	55, 78	nfel 2913	. . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . 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 3513	. . . . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
89	88	fvmpt2 6959	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 = (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥))
92	34	fmpttd 7067	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
93		elpm2r 8792	. . . . . . . . . . . 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 25891	. . . . . . . . . . 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 6842	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
100	99	fvmpt2 6959	. . . . . . . . 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 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 = ⦋0 / 𝑛⦌𝐵)
103	102	oveq1d 7382	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶))
104	103	mpteq2dva 5178	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
105	45, 104	eqtrd 2771	. . . 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 13630	. . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
116		elfznn0 13574	. . . . . . . . 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 25892	. . . . . . 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 7375	. . . . . . 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 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝑗
136		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀))
137	135	nfcsb1 3860	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵
138	137, 78	nfel 2913	. . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . 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 3851	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑛⦌𝐵)
143	142	eleq1d 2821	. . . . . . . . . . . . . . 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 3513	. . . . . . . . . . . . 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 13719	. . . . . . . . . . . . 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 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝑗 + 1)
154		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))
155	153	nfcsb1 3860	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵
156	155, 78	nfel 2913	. . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . 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 3851	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → 𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵)
161	160	eleq1d 2821	. . . . . . . . . . . . . 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 3513	. . . . . . . . . . . 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 2898	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)
169	54, 137	nfmpt 5183	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)
170	168, 169	nfeq 2912	. . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))
174	142	mpteq2dv 5179	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
175	173, 174	eqeq12d 2752	. . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))
180	179	oveq2d 7383	. . . . . . . . . . . 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 25892	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . 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 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1))
189	54, 155	nfmpt 5183	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)
190	188, 189	nfeq 2912	. . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)))
194	160	mpteq2dv 5179	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
195	193, 194	eqeq12d 2752	. . . . . . . . . . . . . . 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 3513	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 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 25932	. . . . . . . . . 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 2775	. . . . . . . 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 2742	. . . . . . 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 2776	. . . . . 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 2775	. . . . 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 13743	. 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 2932  Vcvv 3429  ⦋csb 3837   ⊆ wss 3889  {cpr 4569   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_pm cpm 8774  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   / cdiv 11807  ℕ₀cn0 12437  ℤ_≥cuz 12788  ...cfz 13461  ..^cfzo 13608   D cdv 25830   D^𝑛 cdvn 25831

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-fi 9324  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-icc 13305  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-rest 17385  df-topn 17386  df-0g 17404  df-gsum 17405  df-topgen 17406  df-pt 17407  df-prds 17410  df-xrs 17466  df-qtop 17471  df-imas 17472  df-xps 17474  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-mulg 19044  df-cntz 19292  df-cmn 19757  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-fbas 21349  df-fg 21350  df-cnfld 21353  df-top 22859  df-topon 22876  df-topsp 22898  df-bases 22911  df-cld 22984  df-ntr 22985  df-cls 22986  df-nei 23063  df-lp 23101  df-perf 23102  df-cn 23192  df-cnp 23193  df-haus 23280  df-tx 23527  df-hmeo 23720  df-fil 23811  df-fm 23903  df-flim 23904  df-flf 23905  df-xms 24285  df-ms 24286  df-tms 24287  df-cncf 24845  df-limc 25833  df-dv 25834  df-dvn 25835

This theorem is referenced by: (None)