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Theorem dvfsumabs 25387

Description: Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)

**Hypotheses**
Ref	Expression
dvfsumabs.m	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
dvfsumabs.a	⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℂ))
dvfsumabs.v	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
dvfsumabs.b	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
dvfsumabs.c	⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
dvfsumabs.d	⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
dvfsumabs.x	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
dvfsumabs.y	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
dvfsumabs.l	⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → (abs‘(𝑋 − 𝐵)) ≤ 𝑌)

**Assertion**
Ref	Expression
dvfsumabs	⊢ (𝜑 → (abs‘(Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − (𝐷 − 𝐶))) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑌)

Distinct variable groups: 𝐴,𝑘 𝑥,𝑘,𝑀 𝑘,𝑁,𝑥 𝜑,𝑘,𝑥 𝑥,𝑋 𝑥,𝐶 𝑥,𝐷 𝑥,𝑉 𝑥,𝑌 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥,𝑘) 𝐶(𝑘) 𝐷(𝑘) 𝑉(𝑘) 𝑋(𝑘) 𝑌(𝑘)

Proof of Theorem dvfsumabs Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzofi 13879	. . . . . 6 ⊢ (𝑀..^𝑁) ∈ Fin
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
3		dvfsumabs.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
4		dvfsumabs.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzel2 12768	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		eluzelz 12773	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
8	4, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9		fzval2 13427	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
10	6, 8, 9	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
11		inss1 4188	. . . . . . . . . . 11 ⊢ ((𝑀[,]𝑁) ∩ ℤ) ⊆ (𝑀[,]𝑁)
12	10, 11	eqsstrdi 3998	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀[,]𝑁))
13	12	sselda 3944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → 𝑦 ∈ (𝑀[,]𝑁))
14		dvfsumabs.a	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℂ))
15		cncff 24256	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℂ)
16	14, 15	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℂ)
17		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
18	17	fmpt 7058	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℂ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℂ)
19	16, 18	sylibr 233	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℂ)
20		nfcsb1v 3880	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
21	20	nfel1 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ
22		csbeq1a 3869	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
23	22	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ))
24	21, 23	rspc 3569	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℂ → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ))
25	19, 24	mpan9 507	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
26	13, 25	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
27	26	ralrimiva 3143	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
28		fzofzp1 13669	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
29		csbeq1 3858	. . . . . . . . 9 ⊢ (𝑦 = (𝑘 + 1) → ⦋𝑦 / 𝑥⦌𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
30	29	eleq1d 2822	. . . . . . . 8 ⊢ (𝑦 = (𝑘 + 1) → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ ↔ ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℂ))
31	30	rspccva 3580	. . . . . . 7 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℂ)
32	27, 28, 31	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℂ)
33		elfzofz 13588	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
34		csbeq1 3858	. . . . . . . . 9 ⊢ (𝑦 = 𝑘 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
35	34	eleq1d 2822	. . . . . . . 8 ⊢ (𝑦 = 𝑘 → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ ↔ ⦋𝑘 / 𝑥⦌𝐴 ∈ ℂ))
36	35	rspccva 3580	. . . . . . 7 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ ∧ 𝑘 ∈ (𝑀...𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℂ)
37	27, 33, 36	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℂ)
38	32, 37	subcld 11512	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℂ)
39	2, 3, 38	fsumsub 15673	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) = (Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)))
40		vex 3449	. . . . . . . 8 ⊢ 𝑦 ∈ V
41	40	a1i 11	. . . . . . 7 ⊢ (𝑦 = 𝑀 → 𝑦 ∈ V)
42		eqeq2 2748	. . . . . . . . 9 ⊢ (𝑦 = 𝑀 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑀))
43	42	biimpa 477	. . . . . . . 8 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑀)
44		dvfsumabs.c	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
45	43, 44	syl 17	. . . . . . 7 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐶)
46	41, 45	csbied 3893	. . . . . 6 ⊢ (𝑦 = 𝑀 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐶)
47	40	a1i 11	. . . . . . 7 ⊢ (𝑦 = 𝑁 → 𝑦 ∈ V)
48		eqeq2 2748	. . . . . . . . 9 ⊢ (𝑦 = 𝑁 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑁))
49	48	biimpa 477	. . . . . . . 8 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑁)
50		dvfsumabs.d	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
51	49, 50	syl 17	. . . . . . 7 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐷)
52	47, 51	csbied 3893	. . . . . 6 ⊢ (𝑦 = 𝑁 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐷)
53	34, 29, 46, 52, 4, 26	telfsumo2 15688	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) = (𝐷 − 𝐶))
54	53	oveq2d 7373	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) = (Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − (𝐷 − 𝐶)))
55	39, 54	eqtrd 2776	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) = (Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − (𝐷 − 𝐶)))
56	55	fveq2d 6846	. 2 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) = (abs‘(Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − (𝐷 − 𝐶))))
57	3, 38	subcld 11512	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) ∈ ℂ)
58	2, 57	fsumcl 15618	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) ∈ ℂ)
59	58	abscld 15321	. . 3 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ∈ ℝ)
60	57	abscld 15321	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ∈ ℝ)
61	2, 60	fsumrecl 15619	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(abs‘(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ∈ ℝ)
62		dvfsumabs.y	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
63	2, 62	fsumrecl 15619	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑌 ∈ ℝ)
64	2, 57	fsumabs 15686	. . 3 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ≤ Σ𝑘 ∈ (𝑀..^𝑁)(abs‘(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))))
65		elfzoelz 13572	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
66	65	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℤ)
67	66	zred 12607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ)
68	67	rexrd 11205	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ^*)
69		peano2re 11328	. . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
70	67, 69	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ)
71	70	rexrd 11205	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ^*)
72	67	lep1d 12086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ≤ (𝑘 + 1))
73		ubicc2 13382	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
74	68, 71, 72, 73	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
75		lbicc2 13381	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
76	68, 71, 72, 75	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
77	6	zred 12607	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℝ)
78	77	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
79	8	zred 12607	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℝ)
80	79	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
81		elfzole1 13580	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
82	81	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
83	28	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
84		elfzle2 13445	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
85	83, 84	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
86		iccss 13332	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) ∧ (𝑀 ≤ 𝑘 ∧ (𝑘 + 1) ≤ 𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
87	78, 80, 82, 85, 86	syl22anc 837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
88	87	resmptd 5994	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ ((𝑋 · 𝑥) − 𝐴)) ↾ (𝑘[,](𝑘 + 1))) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))
89		eqid 2736	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
90	89	subcn 24229	. . . . . . . . . . . 12 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
91	90	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
92		iccssre 13346	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
93	77, 79, 92	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
94	93	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℝ)
95		ax-resscn 11108	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
96	94, 95	sstrdi 3956	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℂ)
97		ssid 3966	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
98	97	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℂ ⊆ ℂ)
99		cncfmptc 24275	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℂ ∧ (𝑀[,]𝑁) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝑋) ∈ ((𝑀[,]𝑁)–cn→ℂ))
100	3, 96, 98, 99	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝑋) ∈ ((𝑀[,]𝑁)–cn→ℂ))
101		cncfmptid 24276	. . . . . . . . . . . . 13 ⊢ (((𝑀[,]𝑁) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝑥) ∈ ((𝑀[,]𝑁)–cn→ℂ))
102	96, 97, 101	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝑥) ∈ ((𝑀[,]𝑁)–cn→ℂ))
103	100, 102	mulcncf 24810	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ (𝑋 · 𝑥)) ∈ ((𝑀[,]𝑁)–cn→ℂ))
104	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℂ))
105	89, 91, 103, 104	cncfmpt2f 24278	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ ((𝑋 · 𝑥) − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℂ))
106		rescncf 24260	. . . . . . . . . 10 ⊢ ((𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ ((𝑋 · 𝑥) − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℂ) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ ((𝑋 · 𝑥) − 𝐴)) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℂ)))
107	87, 105, 106	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ ((𝑋 · 𝑥) − 𝐴)) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℂ))
108	88, 107	eqeltrrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)) ∈ ((𝑘[,](𝑘 + 1))–cn→ℂ))
109	95	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ⊆ ℂ)
110	87, 94	sstrd 3954	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℝ)
111	87	sselda 3944	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘[,](𝑘 + 1))) → 𝑥 ∈ (𝑀[,]𝑁))
112	3	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑋 ∈ ℂ)
113	96	sselda 3944	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑥 ∈ ℂ)
114	112, 113	mulcld 11175	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑋 · 𝑥) ∈ ℂ)
115	19	r19.21bi 3234	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
116	115	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
117	114, 116	subcld 11512	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((𝑋 · 𝑥) − 𝐴) ∈ ℂ)
118	111, 117	syldan 591	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘[,](𝑘 + 1))) → ((𝑋 · 𝑥) − 𝐴) ∈ ℂ)
119	89	tgioo2 24166	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120		iccntr 24184	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑘[,](𝑘 + 1))) = (𝑘(,)(𝑘 + 1)))
121	67, 70, 120	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((int‘(topGen‘ran (,)))‘(𝑘[,](𝑘 + 1))) = (𝑘(,)(𝑘 + 1)))
122	109, 110, 118, 119, 89, 121	dvmptntr 25335	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = (ℝ D (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))))
123		reelprrecn 11143	. . . . . . . . . . . . 13 ⊢ ℝ ∈ {ℝ, ℂ}
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ∈ {ℝ, ℂ})
125		ioossicc 13350	. . . . . . . . . . . . . 14 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
126	125	sseli 3940	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
127	126, 117	sylan2 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → ((𝑋 · 𝑥) − 𝐴) ∈ ℂ)
128		ovex 7390	. . . . . . . . . . . . 13 ⊢ (𝑋 − 𝐵) ∈ V
129	128	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝑋 − 𝐵) ∈ V)
130	126, 114	sylan2 593	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝑋 · 𝑥) ∈ ℂ)
131	3	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝑋 ∈ ℂ)
132	125, 96	sstrid 3955	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℂ)
133	132	sselda 3944	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝑥 ∈ ℂ)
134		1cnd 11150	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 1 ∈ ℂ)
135	109	sselda 3944	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
136		1cnd 11150	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
137	124	dvmptid 25321	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
138	125, 94	sstrid 3955	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℝ)
139		iooretop 24129	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀(,)𝑁) ∈ (topGen‘ran (,))
140	139	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ∈ (topGen‘ran (,)))
141	124, 135, 136, 137, 138, 119, 89, 140	dvmptres 25327	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝑥)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 1))
142	124, 133, 134, 141, 3	dvmptcmul 25328	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 · 𝑥))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 · 1)))
143	3	mulid1d 11172	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 1) = 𝑋)
144	143	mpteq2dv 5207	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 · 1)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝑋))
145	142, 144	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 · 𝑥))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝑋))
146	126, 116	sylan2 593	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
147		dvfsumabs.v	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
148	147	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
149		dvfsumabs.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
150	149	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
151	124, 130, 131, 145, 146, 148, 150	dvmptsub 25331	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ ((𝑋 · 𝑥) − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 − 𝐵)))
152	78	rexrd 11205	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
153		iooss1 13299	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
154	152, 82, 153	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
155	80	rexrd 11205	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
156		iooss2 13300	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
157	155, 85, 156	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
158	154, 157	sstrd 3954	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
159		iooretop 24129	. . . . . . . . . . . . 13 ⊢ (𝑘(,)(𝑘 + 1)) ∈ (topGen‘ran (,))
160	159	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ∈ (topGen‘ran (,)))
161	124, 127, 129, 151, 158, 119, 89, 160	dvmptres 25327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)))
162	122, 161	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)))
163	162	dmeqd 5861	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = dom (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)))
164		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)) = (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵))
165	128, 164	dmmpti 6645	. . . . . . . . 9 ⊢ dom (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)) = (𝑘(,)(𝑘 + 1))
166	163, 165	eqtrdi 2792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = (𝑘(,)(𝑘 + 1)))
167	162	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → (ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)))
168	167	fveq1d 6844	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → ((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥) = ((𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵))‘𝑥))
169		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑥 ∈ (𝑘(,)(𝑘 + 1)))
170	164	fvmpt2 6959	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (𝑘(,)(𝑘 + 1)) ∧ (𝑋 − 𝐵) ∈ V) → ((𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵))‘𝑥) = (𝑋 − 𝐵))
171	169, 128, 170	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → ((𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵))‘𝑥) = (𝑋 − 𝐵))
172	168, 171	eqtrd 2776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → ((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥) = (𝑋 − 𝐵))
173	172	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → (abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥)) = (abs‘(𝑋 − 𝐵)))
174		dvfsumabs.l	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → (abs‘(𝑋 − 𝐵)) ≤ 𝑌)
175	174	anassrs 468	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → (abs‘(𝑋 − 𝐵)) ≤ 𝑌)
176	173, 175	eqbrtrd 5127	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → (abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥)) ≤ 𝑌)
177	176	ralrimiva 3143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑘(,)(𝑘 + 1))(abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥)) ≤ 𝑌)
178		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥abs
179		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ℝ
180		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 D
181		nfmpt1 5213	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))
182	179, 180, 181	nfov 7387	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))
183		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑦
184	182, 183	nffv 6852	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦)
185	178, 184	nffv 6852	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦))
186		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ≤
187		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑌
188	185, 186, 187	nfbr 5152	. . . . . . . . . 10 ⊢ Ⅎ𝑥(abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦)) ≤ 𝑌
189		2fveq3 6847	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥)) = (abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦)))
190	189	breq1d 5115	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥)) ≤ 𝑌 ↔ (abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦)) ≤ 𝑌))
191	188, 190	rspc 3569	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑘(,)(𝑘 + 1)) → (∀𝑥 ∈ (𝑘(,)(𝑘 + 1))(abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥)) ≤ 𝑌 → (abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦)) ≤ 𝑌))
192	177, 191	mpan9 507	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → (abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦)) ≤ 𝑌)
193	67, 70, 108, 166, 62, 192	dvlip 25357	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ ((𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑘 ∈ (𝑘[,](𝑘 + 1)))) → (abs‘(((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘(𝑘 + 1)) − ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘𝑘))) ≤ (𝑌 · (abs‘((𝑘 + 1) − 𝑘))))
194	193	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (((𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑘 ∈ (𝑘[,](𝑘 + 1))) → (abs‘(((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘(𝑘 + 1)) − ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘𝑘))) ≤ (𝑌 · (abs‘((𝑘 + 1) − 𝑘)))))
195	74, 76, 194	mp2and 697	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘(((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘(𝑘 + 1)) − ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘𝑘))) ≤ (𝑌 · (abs‘((𝑘 + 1) − 𝑘))))
196		ovex 7390	. . . . . . . . 9 ⊢ ((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴) ∈ V
197		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑘 + 1)
198		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑋 · (𝑘 + 1))
199		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 −
200		nfcsb1v 3880	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑘 + 1) / 𝑥⦌𝐴
201	198, 199, 200	nfov 7387	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴)
202		oveq2 7365	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝑋 · 𝑥) = (𝑋 · (𝑘 + 1)))
203		csbeq1a 3869	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → 𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
204	202, 203	oveq12d 7375	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → ((𝑋 · 𝑥) − 𝐴) = ((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴))
205		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))
206	197, 201, 204, 205	fvmptf 6969	. . . . . . . . 9 ⊢ (((𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)) ∧ ((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴) ∈ V) → ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘(𝑘 + 1)) = ((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴))
207	74, 196, 206	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘(𝑘 + 1)) = ((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴))
208	67	recnd 11183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℂ)
209	3, 208	mulcld 11175	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 𝑘) ∈ ℂ)
210	209, 37	subcld 11512	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℂ)
211		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑘
212		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑋 · 𝑘)
213		nfcsb1v 3880	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐴
214	212, 199, 213	nfov 7387	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴)
215		oveq2 7365	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → (𝑋 · 𝑥) = (𝑋 · 𝑘))
216		csbeq1a 3869	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → 𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
217	215, 216	oveq12d 7375	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → ((𝑋 · 𝑥) − 𝐴) = ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴))
218	211, 214, 217, 205	fvmptf 6969	. . . . . . . . 9 ⊢ ((𝑘 ∈ (𝑘[,](𝑘 + 1)) ∧ ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℂ) → ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘𝑘) = ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴))
219	76, 210, 218	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘𝑘) = ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴))
220	207, 219	oveq12d 7375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘(𝑘 + 1)) − ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘𝑘)) = (((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴) − ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴)))
221		peano2cn 11327	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℂ → (𝑘 + 1) ∈ ℂ)
222	208, 221	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℂ)
223	3, 222	mulcld 11175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · (𝑘 + 1)) ∈ ℂ)
224	223, 209, 32, 37	sub4d 11561	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) = (((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴) − ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴)))
225		1cnd 11150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
226	208, 225	pncan2d 11514	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑘 + 1) − 𝑘) = 1)
227	226	oveq2d 7373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = (𝑋 · 1))
228	3, 222, 208	subdid 11611	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)))
229	227, 228, 143	3eqtr3d 2784	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) = 𝑋)
230	229	oveq1d 7372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) = (𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)))
231	220, 224, 230	3eqtr2rd 2783	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) = (((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘(𝑘 + 1)) − ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘𝑘)))
232	231	fveq2d 6846	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) = (abs‘(((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘(𝑘 + 1)) − ((𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))‘𝑘))))
233	226	fveq2d 6846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘((𝑘 + 1) − 𝑘)) = (abs‘1))
234		abs1 15182	. . . . . . . 8 ⊢ (abs‘1) = 1
235	233, 234	eqtrdi 2792	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘((𝑘 + 1) − 𝑘)) = 1)
236	235	oveq2d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑌 · (abs‘((𝑘 + 1) − 𝑘))) = (𝑌 · 1))
237	62	recnd 11183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℂ)
238	237	mulid1d 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑌 · 1) = 𝑌)
239	236, 238	eqtr2d 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑌 = (𝑌 · (abs‘((𝑘 + 1) − 𝑘))))
240	195, 232, 239	3brtr4d 5137	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ≤ 𝑌)
241	2, 60, 62, 240	fsumle 15684	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(abs‘(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑌)
242	59, 61, 63, 64, 241	letrd 11312	. 2 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑌)
243	56, 242	eqbrtrrd 5129	1 ⊢ (𝜑 → (abs‘(Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − (𝐷 − 𝐶))) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 ⦋csb 3855 ∩ cin 3909 ⊆ wss 3910 {cpr 4588 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ran crn 5634 ↾ cres 5635 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 1c1 11052 + caddc 11054 · cmul 11056 ℝ^*cxr 11188 ≤ cle 11190 − cmin 11385 ℤcz 12499 ℤ_≥cuz 12763 (,)cioo 13264 [,]cicc 13267 ...cfz 13424 ..^cfzo 13567 abscabs 15119 Σcsu 15570 TopOpenctopn 17303 topGenctg 17319 ℂ_fldccnfld 20796 intcnt 22368 Cn ccn 22575 ×_t ctx 22911 –cn→ccncf 24239 D cdv 25227

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231

This theorem is referenced by: (None)

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