mettrifi - Mathbox for Jeff Madsen

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Theorem mettrifi 37081

Description: Generalized triangle inequality for arbitrary finite sums. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)

**Hypotheses**
Ref	Expression
mettrifi.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
mettrifi.3	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
mettrifi.4	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑋)

**Assertion**
Ref	Expression
mettrifi	⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))

Distinct variable groups: 𝐷,𝑘 𝑘,𝐹 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘 𝑘,𝑋

Proof of Theorem mettrifi Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mettrifi.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		eluzfz2 13505	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2813	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
6	5	oveq2d 7417	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑀)))
7		oveq1 7408	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
8	7	oveq2d 7417	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
9	8	sumeq1d 15643	. . . . . . 7 ⊢ (𝑥 = 𝑀 → Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
10	6, 9	breq12d 5151	. . . . . 6 ⊢ (𝑥 = 𝑀 → (((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
11	4, 10	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) ↔ (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
12	11	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))))
13		eleq1 2813	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
14		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
15	14	oveq2d 7417	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑛)))
16		oveq1 7408	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑥 − 1) = (𝑛 − 1))
17	16	oveq2d 7417	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑛 − 1)))
18	17	sumeq1d 15643	. . . . . . 7 ⊢ (𝑥 = 𝑛 → Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
19	15, 18	breq12d 5151	. . . . . 6 ⊢ (𝑥 = 𝑛 → (((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
20	13, 19	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) ↔ (𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
21	20	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))))
22		eleq1 2813	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
23		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
24	23	oveq2d 7417	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))))
25		oveq1 7408	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 1) = ((𝑛 + 1) − 1))
26	25	oveq2d 7417	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑛 + 1) − 1)))
27	26	sumeq1d 15643	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
28	24, 27	breq12d 5151	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
29	22, 28	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
30	29	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))))
31		eleq1 2813	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
32		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁))
33	32	oveq2d 7417	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑁)))
34		oveq1 7408	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥 − 1) = (𝑁 − 1))
35	34	oveq2d 7417	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑁 − 1)))
36	35	sumeq1d 15643	. . . . . . 7 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
37	33, 36	breq12d 5151	. . . . . 6 ⊢ (𝑥 = 𝑁 → (((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
38	31, 37	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) ↔ (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
39	38	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) ≤ Σ𝑘 ∈ (𝑀...(𝑥 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))))
40		0le0 12309	. . . . . . . 8 ⊢ 0 ≤ 0
41	40	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
42		mettrifi.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
43		eluzfz1 13504	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
44	1, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
45		mettrifi.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑋)
46	45	ralrimiva 3138	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋)
47		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
48	47	eleq1d 2810	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑀) ∈ 𝑋))
49	48	rspcv 3600	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋 → (𝐹‘𝑀) ∈ 𝑋))
50	44, 46, 49	sylc 65	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑋)
51		met0 24170	. . . . . . . 8 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) = 0)
52	42, 50, 51	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) = 0)
53		eluzel2 12823	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
54	1, 53	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
55	54	zred 12662	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
56	55	ltm1d 12142	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
57		peano2zm 12601	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
58		fzn 13513	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
59	54, 57, 58	syl2anc2 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
60	56, 59	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
61	60	sumeq1d 15643	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ ∅ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
62		sum0 15663	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = 0
63	61, 62	eqtrdi 2780	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = 0)
64	41, 52, 63	3brtr4d 5170	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
65	64	a1d 25	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
66	65	a1i 11	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
67		peano2fzr 13510	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
68	67	ex 412	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
69	68	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
70	69	imim1d 82	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
71	42	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝐷 ∈ (Met‘𝑋))
72	50	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘𝑀) ∈ 𝑋)
73		simp3 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
74	46	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋)
75		fveq2 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
76	75	eleq1d 2810	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑋))
77	76	rspcv 3600	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋 → (𝐹‘(𝑛 + 1)) ∈ 𝑋))
78	73, 74, 77	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑋)
79		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
80	79	eleq1d 2810	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑛) ∈ 𝑋))
81	80	cbvralvw 3226	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋 ↔ ∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋)
82	74, 81	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋)
83	69	3impia 1114	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
84		rsp 3236	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑛) ∈ 𝑋))
85	82, 83, 84	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘𝑛) ∈ 𝑋)
86		mettri 24179	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ((𝐹‘𝑀) ∈ 𝑋 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑋 ∧ (𝐹‘𝑛) ∈ 𝑋)) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))))
87	71, 72, 78, 85, 86	syl13anc 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))))
88		metcl 24159	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ)
89	71, 72, 78, 88	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ)
90		metcl 24159	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋 ∧ (𝐹‘𝑛) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ∈ ℝ)
91	71, 72, 85, 90	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ∈ ℝ)
92		metcl 24159	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑛) ∈ 𝑋 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑋) → ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ)
93	71, 85, 78, 92	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ)
94	91, 93	readdcld 11239	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∈ ℝ)
95		fzfid 13934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...𝑛) ∈ Fin)
96	71	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐷 ∈ (Met‘𝑋))
97		elfzuz3 13494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑛))
98	83, 97	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑛))
99		fzss2 13537	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
100	98, 99	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
101	100	sselda 3974	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (𝑀...𝑁))
102	45	3ad2antl1 1182	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑋)
103	101, 102	syldan 590	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ 𝑋)
104		elfzuz 13493	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ_≥‘𝑀))
105	104	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (ℤ_≥‘𝑀))
106		peano2uz 12881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
107	105, 106	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
108		elfzuz3 13494	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝑛 + 1)))
109	73, 108	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ_≥‘(𝑛 + 1)))
110	109	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑁 ∈ (ℤ_≥‘(𝑛 + 1)))
111		elfzuz3 13494	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑛 ∈ (ℤ_≥‘𝑘))
112	111	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑛 ∈ (ℤ_≥‘𝑘))
113		eluzp1p1 12846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (ℤ_≥‘𝑘) → (𝑛 + 1) ∈ (ℤ_≥‘(𝑘 + 1)))
114	112, 113	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑛 + 1) ∈ (ℤ_≥‘(𝑘 + 1)))
115		uztrn 12836	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘(𝑛 + 1)) ∧ (𝑛 + 1) ∈ (ℤ_≥‘(𝑘 + 1))) → 𝑁 ∈ (ℤ_≥‘(𝑘 + 1)))
116	110, 114, 115	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑁 ∈ (ℤ_≥‘(𝑘 + 1)))
117		elfzuzb 13491	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) ↔ ((𝑘 + 1) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ (ℤ_≥‘(𝑘 + 1))))
118	107, 116, 117	sylanbrc 582	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑘 + 1) ∈ (𝑀...𝑁))
119		fveq2 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
120	119	eleq1d 2810	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ 𝑋 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑋))
121	120	rspccva 3603	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋)
122	82, 121	sylan 579	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋)
123	118, 122	syldan 590	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋)
124		metcl 24159	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘(𝑘 + 1)) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
125	96, 103, 123, 124	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
126	95, 125	fsumrecl 15676	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
127		letr 11304	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ ∧ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∈ ℝ ∧ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ) → ((((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∧ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
128	89, 94, 126, 127	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∧ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
129	87, 128	mpand 692	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
130		fzfid 13934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...(𝑛 − 1)) ∈ Fin)
131		fzssp1 13540	. . . . . . . . . . . . . . . 16 ⊢ (𝑀...(𝑛 − 1)) ⊆ (𝑀...((𝑛 − 1) + 1))
132		eluzelz 12828	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
133	132	3ad2ant2 1131	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ ℤ)
134	133	zcnd 12663	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ ℂ)
135		ax-1cn 11163	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
136		npcan 11465	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
137	134, 135, 136	sylancl 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝑛 − 1) + 1) = 𝑛)
138	137	oveq2d 7417	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...((𝑛 − 1) + 1)) = (𝑀...𝑛))
139	131, 138	sseqtrid 4026	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...(𝑛 − 1)) ⊆ (𝑀...𝑛))
140	139	sselda 3974	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑛 − 1))) → 𝑘 ∈ (𝑀...𝑛))
141	140, 125	syldan 590	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑛 − 1))) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
142	130, 141	fsumrecl 15676	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
143	91, 142, 93	leadd1d 11804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ (Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))))))
144		simp2 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (ℤ_≥‘𝑀))
145	125	recnd 11238	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℂ)
146		fvoveq1 7424	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
147	79, 146	oveq12d 7419	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
148	144, 145, 147	fsumm1 15693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = (Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))))
149	148	breq2d 5150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ (Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))))))
150	143, 149	bitr4d 282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
151		pncan 11462	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
152	134, 135, 151	sylancl 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝑛 + 1) − 1) = 𝑛)
153	152	oveq2d 7417	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...((𝑛 + 1) − 1)) = (𝑀...𝑛))
154	153	sumeq1d 15643	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
155	154	breq2d 5150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ↔ ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
156	129, 150, 155	3imtr4d 294	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
157	156	3expia 1118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
158	157	a2d 29	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
159	70, 158	syld 47	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
160	159	expcom 413	. . . . 5 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝜑 → ((𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))))
161	160	a2d 29	. . . 4 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ≤ Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ≤ Σ𝑘 ∈ (𝑀...((𝑛 + 1) − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))))
162	12, 21, 30, 39, 66, 161	uzind4 12886	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
163	1, 162	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
164	3, 163	mpd 15	1 ⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⊆ wss 3940 ∅c0 4314 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 < clt 11244 ≤ cle 11245 − cmin 11440 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 Σcsu 15628 Metcmet 21213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-xadd 13089 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-xmet 21220 df-met 21221

This theorem is referenced by: geomcau 37083

W3C validator