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Theorem dvfsumabs 24303

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 13192	. . . . . 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 12098	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		eluzelz 12103	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
8	4, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9		fzval2 12745	. . . . . . . . . . . 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 4125	. . . . . . . . . . 11 ⊢ ((𝑀[,]𝑁) ∩ ℤ) ⊆ (𝑀[,]𝑁)
12	10, 11	syl6eqss 3942	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀[,]𝑁))
13	12	sselda 3889	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → 𝑦 ∈ (𝑀[,]𝑁))
14		dvfsumabs.a	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℂ))
15		cncff 23184	. . . . . . . . . . . 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 2795	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
18	17	fmpt 6737	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℂ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℂ)
19	16, 18	sylibr 235	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℂ)
20		nfcsb1v 3833	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
21	20	nfel1 2963	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ
22		csbeq1a 3824	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
23	22	eleq1d 2867	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ))
24	21, 23	rspc 3553	. . . . . . . . . 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 3149	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
28		fzofzp1 12984	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
29		csbeq1 3814	. . . . . . . . 9 ⊢ (𝑦 = (𝑘 + 1) → ⦋𝑦 / 𝑥⦌𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
30	29	eleq1d 2867	. . . . . . . 8 ⊢ (𝑦 = (𝑘 + 1) → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ ↔ ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℂ))
31	30	rspccva 3558	. . . . . . 7 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℂ)
32	27, 28, 31	syl2an 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℂ)
33		elfzofz 12903	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
34		csbeq1 3814	. . . . . . . . 9 ⊢ (𝑦 = 𝑘 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
35	34	eleq1d 2867	. . . . . . . 8 ⊢ (𝑦 = 𝑘 → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ ↔ ⦋𝑘 / 𝑥⦌𝐴 ∈ ℂ))
36	35	rspccva 3558	. . . . . . 7 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ ∧ 𝑘 ∈ (𝑀...𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℂ)
37	27, 33, 36	syl2an 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℂ)
38	32, 37	subcld 10845	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℂ)
39	2, 3, 38	fsumsub 14976	. . . 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 3440	. . . . . . . 8 ⊢ 𝑦 ∈ V
41	40	a1i 11	. . . . . . 7 ⊢ (𝑦 = 𝑀 → 𝑦 ∈ V)
42		eqeq2 2806	. . . . . . . . 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 3844	. . . . . 6 ⊢ (𝑦 = 𝑀 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐶)
47	40	a1i 11	. . . . . . 7 ⊢ (𝑦 = 𝑁 → 𝑦 ∈ V)
48		eqeq2 2806	. . . . . . . . 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 3844	. . . . . 6 ⊢ (𝑦 = 𝑁 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐷)
53	34, 29, 46, 52, 4, 26	telfsumo2 14991	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) = (𝐷 − 𝐶))
54	53	oveq2d 7032	. . . 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 2831	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) = (Σ𝑘 ∈ (𝑀..^𝑁)𝑋 − (𝐷 − 𝐶)))
56	55	fveq2d 6542	. 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 10845	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) ∈ ℂ)
58	2, 57	fsumcl 14923	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴)) ∈ ℂ)
59	58	abscld 14630	. . 3 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ∈ ℝ)
60	57	abscld 14630	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ∈ ℝ)
61	2, 60	fsumrecl 14924	. . 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 14924	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑌 ∈ ℝ)
64	2, 57	fsumabs 14989	. . 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 12888	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
66	65	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℤ)
67	66	zred 11936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ)
68	67	rexrd 10537	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ^*)
69		peano2re 10660	. . . . . . . . 9 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
70	67, 69	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ)
71	70	rexrd 10537	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ^*)
72	67	lep1d 11419	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ≤ (𝑘 + 1))
73		ubicc2 12703	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
74	68, 71, 72, 73	syl3anc 1364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
75		lbicc2 12702	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
76	68, 71, 72, 75	syl3anc 1364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
77	6	zred 11936	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℝ)
78	77	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
79	8	zred 11936	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℝ)
80	79	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
81		elfzole1 12896	. . . . . . . . . . . 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 12761	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
85	83, 84	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
86		iccss 12654	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) ∧ (𝑀 ≤ 𝑘 ∧ (𝑘 + 1) ≤ 𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
87	78, 80, 82, 85, 86	syl22anc 835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
88	87	resmptd 5789	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ ((𝑋 · 𝑥) − 𝐴)) ↾ (𝑘[,](𝑘 + 1))) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))
89		eqid 2795	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
90	89	subcn 23157	. . . . . . . . . . . 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 12668	. . . . . . . . . . . . . . . 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 10440	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
96	94, 95	syl6ss 3901	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℂ)
97		ssid 3910	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
98	97	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℂ ⊆ ℂ)
99		cncfmptc 23202	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℂ ∧ (𝑀[,]𝑁) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝑋) ∈ ((𝑀[,]𝑁)–cn→ℂ))
100	3, 96, 98, 99	syl3anc 1364	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝑋) ∈ ((𝑀[,]𝑁)–cn→ℂ))
101		cncfmptid 23203	. . . . . . . . . . . . 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 23730	. . . . . . . . . . 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 23205	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ ((𝑋 · 𝑥) − 𝐴)) ∈ ((𝑀[,]𝑁)–cn→ℂ))
106		rescncf 23188	. . . . . . . . . 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 2884	. . . . . . . 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 3899	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℝ)
111	87	sselda 3889	. . . . . . . . . . . . 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 3889	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑥 ∈ ℂ)
114	112, 113	mulcld 10507	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑋 · 𝑥) ∈ ℂ)
115	19	r19.21bi 3175	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
116	115	adantlr 711	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℂ)
117	114, 116	subcld 10845	. . . . . . . . . . . . 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 23094	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
120		iccntr 23112	. . . . . . . . . . . . 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 24251	. . . . . . . . . . 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 10475	. . . . . . . . . . . . 13 ⊢ ℝ ∈ {ℝ, ℂ}
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ∈ {ℝ, ℂ})
125		ioossicc 12672	. . . . . . . . . . . . . 14 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
126	125	sseli 3885	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
127	126, 117	sylan2 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → ((𝑋 · 𝑥) − 𝐴) ∈ ℂ)
128		ovex 7048	. . . . . . . . . . . . 13 ⊢ (𝑋 − 𝐵) ∈ V
129	128	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → (𝑋 − 𝐵) ∈ V)
130	126, 114	sylan2 592	. . . . . . . . . . . . 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	syl5ss 3900	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℂ)
133	132	sselda 3889	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝑥 ∈ ℂ)
134		1cnd 10482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 1 ∈ ℂ)
135	109	sselda 3889	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
136		1cnd 10482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
137	124	dvmptid 24237	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
138	125, 94	syl5ss 3900	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℝ)
139		iooretop 23057	. . . . . . . . . . . . . . . . 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 24243	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝑥)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 1))
142	124, 133, 134, 141, 3	dvmptcmul 24244	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 · 𝑥))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 · 1)))
143	3	mulid1d 10504	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 1) = 𝑋)
144	143	mpteq2dv 5056	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 · 1)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝑋))
145	142, 144	eqtrd 2831	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 · 𝑥))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝑋))
146	126, 116	sylan2 592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℂ)
147		dvfsumabs.v	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
148	147	adantlr 711	. . . . . . . . . . . . 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 24247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ ((𝑋 · 𝑥) − 𝐴))) = (𝑥 ∈ (𝑀(,)𝑁) ↦ (𝑋 − 𝐵)))
152	78	rexrd 10537	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
153		iooss1 12623	. . . . . . . . . . . . . 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 10537	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
156		iooss2 12624	. . . . . . . . . . . . . 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 3899	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
159		iooretop 23057	. . . . . . . . . . . . 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 24243	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)))
162	122, 161	eqtrd 2831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)))
163	162	dmeqd 5660	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))) = dom (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)))
164		eqid 2795	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)) = (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵))
165	128, 164	dmmpti 6360	. . . . . . . . 9 ⊢ dom (𝑥 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 − 𝐵)) = (𝑘(,)(𝑘 + 1))
166	163, 165	syl6eq 2847	. . . . . . . 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 6540	. . . . . . . . . . . . 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 6645	. . . . . . . . . . . . . 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 2831	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → ((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥) = (𝑋 − 𝐵))
173	172	fveq2d 6542	. . . . . . . . . . 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 4984	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → (abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥)) ≤ 𝑌)
177	176	ralrimiva 3149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑘(,)(𝑘 + 1))(abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑥)) ≤ 𝑌)
178		nfcv 2949	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥abs
179		nfcv 2949	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥ℝ
180		nfcv 2949	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 D
181		nfmpt1 5058	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))
182	179, 180, 181	nfov 7046	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))
183		nfcv 2949	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝑦
184	182, 183	nffv 6548	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦)
185	178, 184	nffv 6548	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦))
186		nfcv 2949	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ≤
187		nfcv 2949	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑌
188	185, 186, 187	nfbr 5009	. . . . . . . . . 10 ⊢ Ⅎ𝑥(abs‘((ℝ D (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)))‘𝑦)) ≤ 𝑌
189		2fveq3 6543	. . . . . . . . . . 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 4972	. . . . . . . . . 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 3553	. . . . . . . . 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 24273	. . . . . . 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 695	. . . . 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 7048	. . . . . . . . 9 ⊢ ((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴) ∈ V
197		nfcv 2949	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑘 + 1)
198		nfcv 2949	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑋 · (𝑘 + 1))
199		nfcv 2949	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 −
200		nfcsb1v 3833	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑘 + 1) / 𝑥⦌𝐴
201	198, 199, 200	nfov 7046	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴)
202		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝑋 · 𝑥) = (𝑋 · (𝑘 + 1)))
203		csbeq1a 3824	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → 𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
204	202, 203	oveq12d 7034	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → ((𝑋 · 𝑥) − 𝐴) = ((𝑋 · (𝑘 + 1)) − ⦋(𝑘 + 1) / 𝑥⦌𝐴))
205		eqid 2795	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴)) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ ((𝑋 · 𝑥) − 𝐴))
206	197, 201, 204, 205	fvmptf 6655	. . . . . . . . 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 10515	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℂ)
209	3, 208	mulcld 10507	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 𝑘) ∈ ℂ)
210	209, 37	subcld 10845	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℂ)
211		nfcv 2949	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑘
212		nfcv 2949	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑋 · 𝑘)
213		nfcsb1v 3833	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐴
214	212, 199, 213	nfov 7046	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴)
215		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → (𝑋 · 𝑥) = (𝑋 · 𝑘))
216		csbeq1a 3824	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → 𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
217	215, 216	oveq12d 7034	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → ((𝑋 · 𝑥) − 𝐴) = ((𝑋 · 𝑘) − ⦋𝑘 / 𝑥⦌𝐴))
218	211, 214, 217, 205	fvmptf 6655	. . . . . . . . 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 7034	. . . . . . 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 10659	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℂ → (𝑘 + 1) ∈ ℂ)
222	208, 221	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℂ)
223	3, 222	mulcld 10507	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · (𝑘 + 1)) ∈ ℂ)
224	223, 209, 32, 37	sub4d 10894	. . . . . . 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 10482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
226	208, 225	pncan2d 10847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑘 + 1) − 𝑘) = 1)
227	226	oveq2d 7032	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = (𝑋 · 1))
228	3, 222, 208	subdid 10944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)))
229	227, 228, 143	3eqtr3d 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) = 𝑋)
230	229	oveq1d 7031	. . . . . . 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 2838	. . . . . 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 6542	. . . . 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 6542	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘((𝑘 + 1) − 𝑘)) = (abs‘1))
234		abs1 14491	. . . . . . . 8 ⊢ (abs‘1) = 1
235	233, 234	syl6eq 2847	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘((𝑘 + 1) − 𝑘)) = 1)
236	235	oveq2d 7032	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑌 · (abs‘((𝑘 + 1) − 𝑘))) = (𝑌 · 1))
237	62	recnd 10515	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℂ)
238	237	mulid1d 10504	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑌 · 1) = 𝑌)
239	236, 238	eqtr2d 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑌 = (𝑌 · (abs‘((𝑘 + 1) − 𝑘))))
240	195, 232, 239	3brtr4d 4994	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (abs‘(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ≤ 𝑌)
241	2, 60, 62, 240	fsumle 14987	. . 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 10644	. 2 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (𝑀..^𝑁)(𝑋 − (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))) ≤ Σ𝑘 ∈ (𝑀..^𝑁)𝑌)
243	56, 242	eqbrtrrd 4986	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 1522 ∈ wcel 2081 ∀wral 3105 Vcvv 3437 ⦋csb 3811 ∩ cin 3858 ⊆ wss 3859 {cpr 4474 class class class wbr 4962 ↦ cmpt 5041 dom cdm 5443 ran crn 5444 ↾ cres 5445 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 Fincfn 8357 ℂcc 10381 ℝcr 10382 1c1 10384 + caddc 10386 · cmul 10388 ℝ^*cxr 10520 ≤ cle 10522 − cmin 10717 ℤcz 11829 ℤ_≥cuz 12093 (,)cioo 12588 [,]cicc 12591 ...cfz 12742 ..^cfzo 12883 abscabs 14427 Σcsu 14876 TopOpenctopn 16524 topGenctg 16540 ℂ_fldccnfld 20227 intcnt 21309 Cn ccn 21516 ×_t ctx 21852 –cn→ccncf 23167 D cdv 24144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679 df-sum 14877 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-cmp 21679 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 df-limc 24147 df-dv 24148

This theorem is referenced by: (None)

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