mettrifi - Mathbox for Jeff Madsen

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Theorem mettrifi 33553

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 12349	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		eleq1 2689	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁)))
5		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
6	5	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑀)))
7		oveq1 6657	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑥 − 1) = (𝑀 − 1))
8	7	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑀 − 1)))
9	8	sumeq1d 14431	. . . . . . 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 4666	. . . . . 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 334	. . . . 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 330	. . . 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 2689	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁)))
14		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛))
15	14	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑛)))
16		oveq1 6657	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑥 − 1) = (𝑛 − 1))
17	16	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑛 − 1)))
18	17	sumeq1d 14431	. . . . . . 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 4666	. . . . . 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 334	. . . . 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 330	. . . 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 2689	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁)))
23		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1)))
24	23	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))))
25		oveq1 6657	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 − 1) = ((𝑛 + 1) − 1))
26	25	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝑀...(𝑥 − 1)) = (𝑀...((𝑛 + 1) − 1)))
27	26	sumeq1d 14431	. . . . . . 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 4666	. . . . . 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 334	. . . . 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 330	. . . 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 2689	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁)))
32		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁))
33	32	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀)𝐷(𝐹‘𝑥)) = ((𝐹‘𝑀)𝐷(𝐹‘𝑁)))
34		oveq1 6657	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥 − 1) = (𝑁 − 1))
35	34	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 − 1)) = (𝑀...(𝑁 − 1)))
36	35	sumeq1d 14431	. . . . . . 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 4666	. . . . . 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 334	. . . . 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 330	. . . 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 11110	. . . . . . . 8 ⊢ 0 ≤ 0
41	40	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
42		mettrifi.2	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
43		eluzfz1 12348	. . . . . . . . . 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 2966	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋)
47		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
48	47	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑀) ∈ 𝑋))
49	48	rspcv 3305	. . . . . . . . 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 22148	. . . . . . . 8 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) = 0)
52	42, 50, 51	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) = 0)
53		eluzel2 11692	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
54	1, 53	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
55	54	zred 11482	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
56	55	ltm1d 10956	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
57		peano2zm 11420	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
58	54, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
59		fzn 12357	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
60	54, 58, 59	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
61	56, 60	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
62	61	sumeq1d 14431	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = Σ𝑘 ∈ ∅ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
63		sum0 14452	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = 0
64	62, 63	syl6eq 2672	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = 0)
65	41, 52, 64	3brtr4d 4685	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))
66	65	a1d 25	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
67	66	a1i 11	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → (𝑀 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑀)) ≤ Σ𝑘 ∈ (𝑀...(𝑀 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
68		peano2fzr 12354	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
69	68	ex 450	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
70	69	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁)))
71	70	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))))))
72	42	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝐷 ∈ (Met‘𝑋))
73	50	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘𝑀) ∈ 𝑋)
74		simp3 1063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑛 + 1) ∈ (𝑀...𝑁))
75	46	3ad2ant1 1082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋)
76		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
77	76	eleq1d 2686	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘(𝑛 + 1)) ∈ 𝑋))
78	77	rspcv 3305	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋 → (𝐹‘(𝑛 + 1)) ∈ 𝑋))
79	74, 75, 78	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) ∈ 𝑋)
80		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
81	80	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑛) ∈ 𝑋))
82	81	cbvralv 3171	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ 𝑋 ↔ ∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋)
83	75, 82	sylib 208	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋)
84	70	3impia 1261	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁))
85		rsp 2929	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑛) ∈ 𝑋))
86	83, 84, 85	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘𝑛) ∈ 𝑋)
87		mettri 22157	. . . . . . . . . . . 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)))))
88	72, 73, 79, 86, 87	syl13anc 1328	. . . . . . . . . . 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)))))
89		metcl 22137	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ)
90	72, 73, 79, 89	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑀)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ)
91		metcl 22137	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑀) ∈ 𝑋 ∧ (𝐹‘𝑛) ∈ 𝑋) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ∈ ℝ)
92	72, 73, 86, 91	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑀)𝐷(𝐹‘𝑛)) ∈ ℝ)
93		metcl 22137	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑛) ∈ 𝑋 ∧ (𝐹‘(𝑛 + 1)) ∈ 𝑋) → ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ)
94	72, 86, 79, 93	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))) ∈ ℝ)
95	92, 94	readdcld 10069	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (((𝐹‘𝑀)𝐷(𝐹‘𝑛)) + ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) ∈ ℝ)
96		fzfid 12772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...𝑛) ∈ Fin)
97	72	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝐷 ∈ (Met‘𝑋))
98		elfzuz3 12339	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑛))
99	84, 98	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑛))
100		fzss2 12381	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
101	99, 100	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...𝑛) ⊆ (𝑀...𝑁))
102	101	sselda 3603	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (𝑀...𝑁))
103	45	3ad2antl1 1223	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑋)
104	102, 103	syldan 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ 𝑋)
105		elfzuz 12338	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ_≥‘𝑀))
106	105	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (ℤ_≥‘𝑀))
107		peano2uz 11741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
108	106, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
109		elfzuz3 12339	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘(𝑛 + 1)))
110	74, 109	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑁 ∈ (ℤ_≥‘(𝑛 + 1)))
111	110	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑁 ∈ (ℤ_≥‘(𝑛 + 1)))
112		elfzuz3 12339	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑛 ∈ (ℤ_≥‘𝑘))
113	112	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑛 ∈ (ℤ_≥‘𝑘))
114		eluzp1p1 11713	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (ℤ_≥‘𝑘) → (𝑛 + 1) ∈ (ℤ_≥‘(𝑘 + 1)))
115	113, 114	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑛 + 1) ∈ (ℤ_≥‘(𝑘 + 1)))
116		uztrn 11704	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘(𝑛 + 1)) ∧ (𝑛 + 1) ∈ (ℤ_≥‘(𝑘 + 1))) → 𝑁 ∈ (ℤ_≥‘(𝑘 + 1)))
117	111, 115, 116	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑁 ∈ (ℤ_≥‘(𝑘 + 1)))
118		elfzuzb 12336	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) ↔ ((𝑘 + 1) ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ (ℤ_≥‘(𝑘 + 1))))
119	108, 117, 118	sylanbrc 698	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝑘 + 1) ∈ (𝑀...𝑁))
120		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
121	120	eleq1d 2686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ 𝑋 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑋))
122	121	rspccva 3308	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) ∈ 𝑋 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋)
123	83, 122	sylan 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋)
124	119, 123	syldan 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘(𝑘 + 1)) ∈ 𝑋)
125		metcl 22137	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘(𝑘 + 1)) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
126	97, 104, 124, 125	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
127	96, 126	fsumrecl 14465	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...𝑛)((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
128		letr 10131	. . . . . . . . . . . 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)))))
129	90, 95, 127, 128	syl3anc 1326	. . . . . . . . . . 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)))))
130	88, 129	mpand 711	. . . . . . . . . 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)))))
131		fzfid 12772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...(𝑛 − 1)) ∈ Fin)
132		fzssp1 12384	. . . . . . . . . . . . . . . 16 ⊢ (𝑀...(𝑛 − 1)) ⊆ (𝑀...((𝑛 − 1) + 1))
133		eluzelz 11697	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
134	133	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ ℤ)
135	134	zcnd 11483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ ℂ)
136		ax-1cn 9994	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
137		npcan 10290	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
138	135, 136, 137	sylancl 694	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝑛 − 1) + 1) = 𝑛)
139	138	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...((𝑛 − 1) + 1)) = (𝑀...𝑛))
140	132, 139	syl5sseq 3653	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...(𝑛 − 1)) ⊆ (𝑀...𝑛))
141	140	sselda 3603	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑛 − 1))) → 𝑘 ∈ (𝑀...𝑛))
142	141, 126	syldan 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...(𝑛 − 1))) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
143	131, 142	fsumrecl 14465	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → Σ𝑘 ∈ (𝑀...(𝑛 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℝ)
144	92, 143, 94	leadd1d 10621	. . . . . . . . . . 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))))))
145		simp2 1062	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (ℤ_≥‘𝑀))
146	126	recnd 10068	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ∈ ℂ)
147		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
148	147	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
149	80, 148	oveq12d 6668	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) = ((𝐹‘𝑛)𝐷(𝐹‘(𝑛 + 1))))
150	145, 146, 149	fsumm1 14480	. . . . . . . . . . . 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)))))
151	150	breq2d 4665	. . . . . . . . . . 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))))))
152	144, 151	bitr4d 271	. . . . . . . . . 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)))))
153		pncan 10287	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
154	135, 136, 153	sylancl 694	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → ((𝑛 + 1) − 1) = 𝑛)
155	154	oveq2d 6666	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝑀...((𝑛 + 1) − 1)) = (𝑀...𝑛))
156	155	sumeq1d 14431	. . . . . . . . . . 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))))
157	156	breq2d 4665	. . . . . . . . . 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)))))
158	130, 152, 157	3imtr4d 283	. . . . . . . . 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)))))
159	158	3expia 1267	. . . . . . . 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))))))
160	159	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))))))
161	71, 160	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))))))
162	161	expcom 451	. . . . 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)))))))
163	162	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)))))))
164	12, 21, 30, 39, 67, 163	uzind4 11746	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))))))
165	1, 164	mpcom 38	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))))
166	3, 165	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 Σcsu 14416 Metcme 19732

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-xmet 19739 df-met 19740

This theorem is referenced by: geomcau 33555

W3C validator