Theorem dvnmptdivc 38828

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 476	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → 𝑛 ∈ (0...𝑀))
2		simpl 472	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → 𝜑)
3		fveq2 6103	. . . . 5 ⊢ (𝑘 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0))
4		csbeq1 3502	. . . . . . 7 ⊢ (𝑘 = 0 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋0 / 𝑛⦌𝐵)
5	4	oveq1d 6564	. . . . . 6 ⊢ (𝑘 = 0 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶))
6	5	mpteq2dv 4673	. . . . 5 ⊢ (𝑘 = 0 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
7	3, 6	eqeq12d 2625	. . . 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 329	. . 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 6103	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗))
10		csbeq1 3502	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ⦋𝑘 / 𝑛⦌𝐵 = ⦋𝑗 / 𝑛⦌𝐵)
11	10	oveq1d 6564	. . . . . 6 ⊢ (𝑘 = 𝑗 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))
12	11	mpteq2dv 4673	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶)))
13	9, 12	eqeq12d 2625	. . . 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 329	. . 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 6103	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)))
16		csbeq1 3502	. . . . . . 7 ⊢ (𝑘 = (𝑗 + 1) → ⦋𝑘 / 𝑛⦌𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵)
17	16	oveq1d 6564	. . . . . 6 ⊢ (𝑘 = (𝑗 + 1) → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶))
18	17	mpteq2dv 4673	. . . . 5 ⊢ (𝑘 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
19	15, 18	eqeq12d 2625	. . . 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 329	. . 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 6103	. . . . 5 ⊢ (𝑘 = 𝑛 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑘) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛))
22		csbeq1a 3508	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
23	22	equcoms 1934	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑘 / 𝑛⦌𝐵)
24	23	eqcomd 2616	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ⦋𝑘 / 𝑛⦌𝐵 = 𝐵)
25	24	oveq1d 6564	. . . . . 6 ⊢ (𝑘 = 𝑛 → (⦋𝑘 / 𝑛⦌𝐵 / 𝐶) = (𝐵 / 𝐶))
26	25	mpteq2dv 4673	. . . . 5 ⊢ (𝑘 = 𝑛 → (𝑥 ∈ 𝑋 ↦ (⦋𝑘 / 𝑛⦌𝐵 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))
27	21, 26	eqeq12d 2625	. . . 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 329	. . 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 23474	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
31	29, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
32		cnex 9896	. . . . . . . 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 10680	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) ∈ ℂ)
40		eqid 2610	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))
41	39, 40	fmptd 6292	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ)
42		dvnmptdivc.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
43		elpm2r 7761	. . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)):𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
44	33, 29, 41, 42, 43	syl22anc 1319	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
45		dvn0 23493	. . . . . 6 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))
46	31, 44, 45	syl2anc 691	. . . . 5 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))
47		id 22	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝜑)
48		dvnmptdivc.8	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
49		nn0uz 11598	. . . . . . . . . . . . . 14 ⊢ ℕ0 = (ℤ≥‘0)
50	48, 49	syl6eleq 2698	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
51		eluzfz1 12219	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑀))
52	50, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ (0...𝑀))
53		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 0 ∈ (0...𝑀))
54		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)
55		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛𝑋
56		nfcsb1v 3515	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋0 / 𝑛⦌𝐵
57	55, 56	nfmpt 4674	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
58	54, 57	nfeq 2762	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
59	53, 58	nfim 1813	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
60		c0ex 9913	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
61		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝑛 ∈ (0...𝑀) ↔ 0 ∈ (0...𝑀)))
62	61	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 0 ∈ (0...𝑀))))
63		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0))
64		csbeq1a 3508	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 0 → 𝐵 = ⦋0 / 𝑛⦌𝐵)
65	64	mpteq2dv 4673	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 0 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
66	63, 65	eqeq12d 2625	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)))
67	62, 66	imbi12d 333	. . . . . . . . . . . . 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))))
68		dvnmptdivc.dvn	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵))
69	59, 60, 67, 68	vtoclf 3231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
70	47, 52, 69	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵))
71	70	fveq1d 6105	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥))
72	71	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥))
73		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
74		simpl 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝜑)
75	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 0 ∈ (0...𝑀))
76		0re 9919	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
77		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛0
78		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀))
79		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛ℂ
80	56, 79	nfel 2763	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋0 / 𝑛⦌𝐵 ∈ ℂ
81	78, 80	nfim 1813	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)
82	61	3anbi3d 1397	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀))))
83	64	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 0 → (𝐵 ∈ ℂ ↔ ⦋0 / 𝑛⦌𝐵 ∈ ℂ))
84	82, 83	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑛 = 0 → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)))
85		dvnmptdivc.b	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)
86	77, 81, 84, 85	vtoclgf 3237	. . . . . . . . . . . 12 ⊢ (0 ∈ ℝ → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ))
87	76, 86	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 0 ∈ (0...𝑀)) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)
88	74, 73, 75, 87	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 ∈ ℂ)
89		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)
90	89	fvmpt2 6200	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ∧ ⦋0 / 𝑛⦌𝐵 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥) = ⦋0 / 𝑛⦌𝐵)
91	73, 88, 90	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⦋0 / 𝑛⦌𝐵)‘𝑥) = ⦋0 / 𝑛⦌𝐵)
92	72, 91	eqtr2d 2645	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋0 / 𝑛⦌𝐵 = (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥))
93		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
94	34, 93	fmptd 6292	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
95		elpm2r 7761	. . . . . . . . . . . 12 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
96	33, 29, 94, 42, 95	syl22anc 1319	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
97		dvn0 23493	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ 𝐴))
98	31, 96, 97	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0) = (𝑥 ∈ 𝑋 ↦ 𝐴))
99	98	fveq1d 6105	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥))
100	99	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘0)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥))
101	93	fvmpt2 6200	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
102	73, 34, 101	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑥) = 𝐴)
103	92, 100, 102	3eqtrrd 2649	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 = ⦋0 / 𝑛⦌𝐵)
104	103	oveq1d 6564	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 / 𝐶) = (⦋0 / 𝑛⦌𝐵 / 𝐶))
105	104	mpteq2dva 4672	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
106	46, 105	eqtrd 2644	. . . 4 ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶)))
107	106	a1i 11	. . 3 ⊢ (𝑀 ∈ (ℤ≥‘0) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘0) = (𝑥 ∈ 𝑋 ↦ (⦋0 / 𝑛⦌𝐵 / 𝐶))))
108		simp3 1056	. . . . 5 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
109		simp1 1054	. . . . 5 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝑗 ∈ (0..^𝑀))
110		simpr 476	. . . . . . 7 ⊢ (((𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) ∧ 𝜑) → 𝜑)
111		simpl 472	. . . . . . 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))))
112	110, 111	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)))
113	112	3adant1 1072	. . . . 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)))
114	31	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑆 ⊆ ℂ)
115	44	ad2antrr 758	. . . . . . 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))
116		elfzofz 12354	. . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
117		elfznn0 12302	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
118	117	ad2antlr 759	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
119	116, 118	sylanl2 681	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗) = (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) → 𝑗 ∈ ℕ0)
120		dvnp1 23494	. . . . . . 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)))
121	114, 115, 119, 120	syl3anc 1318	. . . . . 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)))
122		oveq2 6557	. . . . . . 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))))
123	122	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))))
124	31	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ)
125	44	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)) ∈ (ℂ ↑pm 𝑆))
126		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ (0...𝑀))
127	126, 117	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ0)
128	116, 127	sylan2 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ ℕ0)
129	124, 125, 128, 120	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑗)))
130	129	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)))
131	29	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ})
132		simplr 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑗 ∈ (0...𝑀))
133	47	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝜑)
134		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
135	133, 134, 132	3jca 1235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)))
136		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝑗
137		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀))
138	136	nfcsb1 3514	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵
139	138, 79	nfel 2763	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ
140	137, 139	nfim 1813	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)
141		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑛 ∈ (0...𝑀) ↔ 𝑗 ∈ (0...𝑀)))
142	141	3anbi3d 1397	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀))))
143		csbeq1a 3508	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑛⦌𝐵)
144	143	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ))
145	142, 144	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)))
146	136, 140, 145, 85	vtoclgf 3237	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...𝑀) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑗 ∈ (0...𝑀)) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ))
147	132, 135, 146	sylc 63	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)
148	116, 147	sylanl2 681	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋𝑗 / 𝑛⦌𝐵 ∈ ℂ)
149		fzofzp1 12431	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
150	149	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝑗 + 1) ∈ (0...𝑀))
151	116, 133	sylanl2 681	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝜑)
152		simpr 476	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
153	151, 152, 150	3jca 1235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)))
154		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝑗 + 1)
155		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))
156	154	nfcsb1 3514	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵
157	156, 79	nfel 2763	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ
158	155, 157	nfim 1813	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)
159		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → (𝑛 ∈ (0...𝑀) ↔ (𝑗 + 1) ∈ (0...𝑀)))
160	159	3anbi3d 1397	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀))))
161		csbeq1a 3508	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → 𝐵 = ⦋(𝑗 + 1) / 𝑛⦌𝐵)
162	161	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑗 + 1) → (𝐵 ∈ ℂ ↔ ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ))
163	160, 162	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑗 + 1) → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)))
164	154, 158, 163, 85	vtoclgf 3237	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ))
165	150, 153, 164	sylc 63	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) ∧ 𝑥 ∈ 𝑋) → ⦋(𝑗 + 1) / 𝑛⦌𝐵 ∈ ℂ)
166		simpl 472	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝜑)
167	116	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
168		nfv 1830	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑗 ∈ (0...𝑀))
169		nfcv 2751	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)
170	55, 138	nfmpt 4674	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)
171	169, 170	nfeq 2762	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)
172	168, 171	nfim 1813	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
173	141	anbi2d 736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝑀))))
174		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))
175	143	mpteq2dv 4673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
176	174, 175	eqeq12d 2625	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)))
177	173, 176	imbi12d 333	. . . . . . . . . . . . . . . 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))))
178	172, 177, 68	chvar 2250	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
179	166, 167, 178	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵))
180	179	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗))
181	180	oveq2d 6565	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)))
182	166, 96	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑pm 𝑆))
183		dvnp1 23494	. . . . . . . . . . . . . 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)))
184	124, 182, 128, 183	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)))
185	184	eqcomd 2616	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑗)) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)))
186	149	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
187	166, 186	jca 553	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)))
188		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))
189		nfcv 2751	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1))
190	55, 156	nfmpt 4674	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)
191	189, 190	nfeq 2762	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)
192	188, 191	nfim 1813	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
193	159	anbi2d 736	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑗 + 1) → ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) ↔ (𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀))))
194		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)))
195	161	mpteq2dv 4673	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑗 + 1) → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
196	194, 195	eqeq12d 2625	. . . . . . . . . . . . . . 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)))
197	193, 196	imbi12d 333	. . . . . . . . . . . . . 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))))
198	154, 192, 197, 68	vtoclgf 3237	. . . . . . . . . . . . 13 ⊢ ((𝑗 + 1) ∈ (0...𝑀) → ((𝜑 ∧ (𝑗 + 1) ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵)))
199	186, 187, 198	sylc 63	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑗 + 1)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
200	181, 185, 199	3eqtrd 2648	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑛⦌𝐵)) = (𝑥 ∈ 𝑋 ↦ ⦋(𝑗 + 1) / 𝑛⦌𝐵))
201	35	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝐶 ∈ ℂ)
202	37	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝐶 ≠ 0)
203	131, 148, 165, 200, 201, 202	dvmptdivc 23534	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ (⦋𝑗 / 𝑛⦌𝐵 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (⦋(𝑗 + 1) / 𝑛⦌𝐵 / 𝐶)))
204	203	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)))
205	130, 123, 204	3eqtrd 2648	. . . . . . . 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)))
206	205	eqcomd 2616	. . . . . . 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)))
207	206, 121, 123	3eqtrrd 2649	. . . . . 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)))
208	121, 123, 207	3eqtrd 2648	. . . . 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)))
209	108, 109, 113, 208	syl21anc 1317	. . . 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)))
210	209	3exp 1256	. . 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)))))
211	8, 14, 20, 28, 107, 210	fzind2 12448	. 2 ⊢ (𝑛 ∈ (0...𝑀) → (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))))
212	1, 2, 211	sylc 63	1 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ⦋csb 3499   ⊆ wss 3540  {cpr 4127   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_pm cpm 7745  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   / cdiv 10563  ℕ₀cn0 11169  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334   D cdv 23433   D^𝑛 cdvn 23434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-dvn 23438

This theorem is referenced by: (None)