dvfsumle - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvfsumle		Structured version Visualization version GIF version

Theorem dvfsumle 25994

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.) Avoid ax-mulf 11118. (Revised by GG, 16-Mar-2025.)

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

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

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

Proof of Theorem dvfsumle Dummy variables 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzofi 13909	. . . 4 ⊢ (𝑀..^𝑁) ∈ Fin
2	1	a1i 11	. . 3 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
3		dvfsumle.x	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
4		dvfsumle.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzel2 12768	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		eluzelz 12773	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
8	4, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9		fzval2 13438	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
10	6, 8, 9	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
11		inss1 4191	. . . . . . . . 9 ⊢ ((𝑀[,]𝑁) ∩ ℤ) ⊆ (𝑀[,]𝑁)
12	10, 11	eqsstrdi 3980	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀[,]𝑁))
13	12	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → 𝑦 ∈ (𝑀[,]𝑁))
14		dvfsumle.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
15		cncff 24854	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
17		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
18	17	fmpt 7064	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
19	16, 18	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
20		nfcsb1v 3875	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
21	20	nfel1 2916	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ
22		csbeq1a 3865	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
23	22	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
24	21, 23	rspc 3566	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
25	19, 24	mpan9 506	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
26	13, 25	syldan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
27	26	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
28		fzofzp1 13692	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
29		csbeq1 3854	. . . . . . 7 ⊢ (𝑦 = (𝑘 + 1) → ⦋𝑦 / 𝑥⦌𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
30	29	eleq1d 2822	. . . . . 6 ⊢ (𝑦 = (𝑘 + 1) → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ))
31	30	rspccva 3577	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
32	27, 28, 31	syl2an 597	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
33		elfzofz 13603	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
34		csbeq1 3854	. . . . . . 7 ⊢ (𝑦 = 𝑘 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
35	34	eleq1d 2822	. . . . . 6 ⊢ (𝑦 = 𝑘 → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ))
36	35	rspccva 3577	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ 𝑘 ∈ (𝑀...𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
37	27, 33, 36	syl2an 597	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
38	32, 37	resubcld 11577	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℝ)
39		elfzoelz 13587	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
40	39	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℤ)
41	40	zred 12608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ)
42	41	recnd 11172	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℂ)
43		ax-1cn 11096	. . . . . . 7 ⊢ 1 ∈ ℂ
44		pncan2 11399	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 𝑘) = 1)
45	42, 43, 44	sylancl 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑘 + 1) − 𝑘) = 1)
46	45	oveq2d 7384	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = (𝑋 · 1))
47	3	recnd 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
48		peano2re 11318	. . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
49	41, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ)
50	49	recnd 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℂ)
51	47, 50, 42	subdid 11605	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)))
52	47	mulridd 11161	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 1) = 𝑋)
53	46, 51, 52	3eqtr3d 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) = 𝑋)
54	47	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘[,](𝑘 + 1))) → 𝑋 ∈ ℂ)
55	41, 49	iccssred 13362	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℝ)
56		ax-resscn 11095	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
57	55, 56	sstrdi 3948	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℂ)
58	57	sselda 3935	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘[,](𝑘 + 1))) → 𝑦 ∈ ℂ)
59		ovmpot 7529	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) = (𝑋 · 𝑦))
60	54, 58, 59	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘[,](𝑘 + 1))) → (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) = (𝑋 · 𝑦))
61	60	eqeq2d 2748	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘[,](𝑘 + 1))) → (𝑧 = (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ↔ 𝑧 = (𝑋 · 𝑦)))
62	61	pm5.32da 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑦 ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑧 = (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦)) ↔ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑧 = (𝑋 · 𝑦))))
63	62	opabbidv 5166	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑧 = (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦))} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑧 = (𝑋 · 𝑦))})
64		df-mpt 5182	. . . . . . 7 ⊢ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋 · 𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑧 = (𝑋 · 𝑦))}
65	63, 64	eqtr4di 2790	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑧 = (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦))} = (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋 · 𝑦)))
66		df-mpt 5182	. . . . . . 7 ⊢ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑧 = (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦))}
67		eqid 2737	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
68	67	mpomulcn 24826	. . . . . . . 8 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
69	56	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ⊆ ℂ)
70		cncfmptc 24873	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ ∧ (𝑘[,](𝑘 + 1)) ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑋) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
71	3, 57, 69, 70	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑋) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
72		cncfmptid 24874	. . . . . . . . 9 ⊢ (((𝑘[,](𝑘 + 1)) ⊆ ℝ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
73	55, 56, 72	sylancl 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
74		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑋 ∈ ℝ)
75	74	recnd 11172	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑋 ∈ ℂ)
76		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
77	76	recnd 11172	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
78	59	eqcomd 2743	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑋 · 𝑦) = (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦))
79	75, 77, 78	syl2anc 585	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑋 · 𝑦) = (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦))
80		remulcl 11123	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑋 · 𝑦) ∈ ℝ)
81	79, 80	eqeltrrd 2838	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ∈ ℝ)
82	67, 68, 71, 73, 56, 81	cncfmpt2ss 24877	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦)) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
83	66, 82	eqeltrrid 2842	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (𝑘[,](𝑘 + 1)) ∧ 𝑧 = (𝑋(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦))} ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
84	65, 83	eqeltrrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋 · 𝑦)) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
85		reelprrecn 11130	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
86	85	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ∈ {ℝ, ℂ})
87	6	zred 12608	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
88	87	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
89	88	rexrd 11194	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
90		elfzole1 13595	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
91	90	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
92		iooss1 13308	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
93	89, 91, 92	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
94	8	zred 12608	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℝ)
95	94	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
96	95	rexrd 11194	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
97	28	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
98		elfzle2 13456	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
99	97, 98	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
100		iooss2 13309	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
101	96, 99, 100	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
102	93, 101	sstrd 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
103		ioossicc 13361	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
104	87, 94	iccssred 13362	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
105	104	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℝ)
106	105, 56	sstrdi 3948	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℂ)
107	103, 106	sstrid 3947	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℂ)
108	102, 107	sstrd 3946	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℂ)
109	108	sselda 3935	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑦 ∈ ℂ)
110		1cnd 11139	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 1 ∈ ℂ)
111	69	sselda 3935	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
112		1cnd 11139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
113	86	dvmptid 25929	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
114		ioossre 13335	. . . . . . . . 9 ⊢ (𝑘(,)(𝑘 + 1)) ⊆ ℝ
115	114	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℝ)
116		tgioo4 24761	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
117		iooretop 24721	. . . . . . . . 9 ⊢ (𝑘(,)(𝑘 + 1)) ∈ (topGen‘ran (,))
118	117	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ∈ (topGen‘ran (,)))
119	86, 111, 112, 113, 115, 116, 67, 118	dvmptres 25935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑦)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 1))
120	86, 109, 110, 119, 47	dvmptcmul 25936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)))
121	52	mpteq2dv 5194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
122	120, 121	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
123		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
124	123, 20, 22	cbvmpt 5202	. . . . . 6 ⊢ (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) = (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)
125		iccss 13342	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) ∧ (𝑀 ≤ 𝑘 ∧ (𝑘 + 1) ≤ 𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
126	88, 95, 91, 99, 125	syl22anc 839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
127	126	resmptd 6007	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴))
128	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
129		rescncf 24858	. . . . . . . 8 ⊢ ((𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ)))
130	126, 128, 129	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
131	127, 130	eqeltrrd 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
132	124, 131	eqeltrrid 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
133	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
134	133, 18	sylibr 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
135	103	sseli 3931	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → 𝑦 ∈ (𝑀[,]𝑁))
136	24	impcom 407	. . . . . . . 8 ⊢ ((∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
137	134, 135, 136	syl2an 597	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
138	137	recnd 11172	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
139	103	sseli 3931	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
140	16	fvmptelcdm 7067	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
141	140	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
142	139, 141	sylan2 594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
143	142	fmpttd 7069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
144		ioossre 13335	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ ℝ
145		dvfre 25923	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ ∧ (𝑀(,)𝑁) ⊆ ℝ) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
146	143, 144, 145	sylancl 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
147		dvfsumle.b	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
148	147	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
149	148	dmeqd 5862	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
150		dvfsumle.v	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
151	150	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
152	151	ralrimiva 3130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
153		dmmptg 6208	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
154	152, 153	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
155	149, 154	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
156	148, 155	feq12d 6658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ))
157	146, 156	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
158		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)
159	158	fmpt 7064	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
160	157, 159	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ)
161		nfcsb1v 3875	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
162	161	nfel1 2916	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ
163		csbeq1a 3865	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
164	163	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
165	162, 164	rspc 3566	. . . . . . 7 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
166	160, 165	mpan9 506	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ)
167	123, 20, 22	cbvmpt 5202	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)
168	167	oveq2i 7379	. . . . . . 7 ⊢ (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴))
169		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
170	169, 161, 163	cbvmpt 5202	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵)
171	148, 168, 170	3eqtr3g 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵))
172	86, 138, 166, 171, 102, 116, 67, 118	dvmptres 25935	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐵))
173		dvfsumle.l	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ≤ 𝐵)
174	173	anassrs 467	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ 𝐵)
175	174	ralrimiva 3130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵)
176		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑋
177		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
178	176, 177, 161	nfbr 5147	. . . . . . 7 ⊢ Ⅎ𝑥 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵
179	163	breq2d 5112	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 ≤ 𝐵 ↔ 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
180	178, 179	rspc 3566	. . . . . 6 ⊢ (𝑦 ∈ (𝑘(,)(𝑘 + 1)) → (∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵 → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
181	175, 180	mpan9 506	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵)
182	41	rexrd 11194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ^*)
183	49	rexrd 11194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ^*)
184	41	lep1d 12085	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ≤ (𝑘 + 1))
185		lbicc2 13392	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
186	182, 183, 184, 185	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
187		ubicc2 13393	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
188	182, 183, 184, 187	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
189		oveq2 7376	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑋 · 𝑦) = (𝑋 · 𝑘))
190		oveq2 7376	. . . . 5 ⊢ (𝑦 = (𝑘 + 1) → (𝑋 · 𝑦) = (𝑋 · (𝑘 + 1)))
191	41, 49, 84, 122, 132, 172, 181, 186, 188, 184, 189, 34, 190, 29	dvle 25980	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
192	53, 191	eqbrtrrd 5124	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
193	2, 3, 38, 192	fsumle 15734	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
194		vex 3446	. . . . 5 ⊢ 𝑦 ∈ V
195	194	a1i 11	. . . 4 ⊢ (𝑦 = 𝑀 → 𝑦 ∈ V)
196		eqeq2 2749	. . . . . 6 ⊢ (𝑦 = 𝑀 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑀))
197	196	biimpa 476	. . . . 5 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑀)
198		dvfsumle.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
199	197, 198	syl 17	. . . 4 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐶)
200	195, 199	csbied 3887	. . 3 ⊢ (𝑦 = 𝑀 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐶)
201	194	a1i 11	. . . 4 ⊢ (𝑦 = 𝑁 → 𝑦 ∈ V)
202		eqeq2 2749	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑁))
203	202	biimpa 476	. . . . 5 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑁)
204		dvfsumle.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
205	203, 204	syl 17	. . . 4 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐷)
206	201, 205	csbied 3887	. . 3 ⊢ (𝑦 = 𝑁 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐷)
207	26	recnd 11172	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
208	34, 29, 200, 206, 4, 207	telfsumo2 15738	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) = (𝐷 − 𝐶))
209	193, 208	breqtrd 5126	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3442 ⦋csb 3851 ∩ cin 3902 ⊆ wss 3903 {cpr 4584 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 dom cdm 5632 ran crn 5633 ↾ cres 5634 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 Fincfn 8895 ℂcc 11036 ℝcr 11037 1c1 11039 + caddc 11041 · cmul 11043 ℝ^*cxr 11177 ≤ cle 11179 − cmin 11376 ℤcz 12500 ℤ_≥cuz 12763 (,)cioo 13273 [,]cicc 13276 ...cfz 13435 ..^cfzo 13582 Σcsu 15621 TopOpenctopn 17353 topGenctg 17369 ℂ_fldccnfld 21321 –cn→ccncf 24837 D cdv 25832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836

This theorem is referenced by: dvfsumge 25996

W3C validator