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Theorem telgsumfzs 19590

Description: Telescoping group sum ranging over a finite set of sequential integers, using explicit substitution. (Contributed by AV, 23-Nov-2019.)

**Hypotheses**
Ref	Expression
telgsumfzs.b	⊢ 𝐵 = (Base‘𝐺)
telgsumfzs.g	⊢ (𝜑 → 𝐺 ∈ Abel)
telgsumfzs.m	⊢ − = (-_g‘𝐺)
telgsumfzs.n	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
telgsumfzs.f	⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵)

**Assertion**
Ref	Expression
telgsumfzs	⊢ (𝜑 → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶))

Distinct variable groups: 𝐵,𝑖,𝑘 𝐶,𝑖 𝑖,𝐺 𝑖,𝑀,𝑘 − ,𝑖 𝜑,𝑖 𝑖,𝑁,𝑘 Allowed substitution hints: 𝜑(𝑘) 𝐶(𝑘) 𝐺(𝑘) − (𝑘)

Proof of Theorem telgsumfzs Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		telgsumfzs.f	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵)
2		telgsumfzs.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
3		oveq1 7282	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1))
4	3	oveq2d 7291	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1)))
5	4	raleqdv 3348	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵))
6	5	anbi2d 629	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵)))
7		oveq2 7283	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
8	7	mpteq1d 5169	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)) = (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))
9	8	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))))
10	3	csbeq1d 3836	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶)
11	10	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
12	9, 11	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) ↔ (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶)))
13	6, 12	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑀 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))))
14		oveq1 7282	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
15	14	oveq2d 7291	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1)))
16	15	raleqdv 3348	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
17	16	anbi2d 629	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)))
18		oveq2 7283	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦))
19	18	mpteq1d 5169	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)) = (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))
20	19	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))))
21	14	csbeq1d 3836	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶)
22	21	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶))
23	20, 22	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) ↔ (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶)))
24	17, 23	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶))))
25		oveq1 7282	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
26	25	oveq2d 7291	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1)))
27	26	raleqdv 3348	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵))
28	27	anbi2d 629	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)))
29		oveq2 7283	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1)))
30	29	mpteq1d 5169	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)) = (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))
31	30	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (𝐺 Σ_g (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))))
32	25	csbeq1d 3836	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)
33	32	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
34	31, 33	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) ↔ (𝐺 Σ_g (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)))
35	28, 34	imbi12d 345	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))))
36		oveq1 7282	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
37	36	oveq2d 7291	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1)))
38	37	raleqdv 3348	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵))
39	38	anbi2d 629	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵)))
40		oveq2 7283	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
41	40	mpteq1d 5169	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))
42	41	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))))
43	36	csbeq1d 3836	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑁 + 1) / 𝑘⦌𝐶)
44	43	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶))
45	42, 44	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) ↔ (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶)))
46	39, 45	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑁 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑥) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶))))
47		eluzel2 12587	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
48	2, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
49	48	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ ℤ)
50		fzsn 13298	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
51	49, 50	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀...𝑀) = {𝑀})
52	51	mpteq1d 5169	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)) = (𝑖 ∈ {𝑀} ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))
53	52	oveq2d 7291	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (𝐺 Σ_g (𝑖 ∈ {𝑀} ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))))
54		telgsumfzs.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
55		telgsumfzs.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ Abel)
56		ablgrp 19391	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
57	55, 56	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp)
58	57	grpmndd 18589	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
59	58	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Mnd)
60	57	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Grp)
61		uzid 12597	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
62	49, 61	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (ℤ_≥‘𝑀))
63		peano2uz 12641	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝑀) → (𝑀 + 1) ∈ (ℤ_≥‘𝑀))
64	62, 63	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀 + 1) ∈ (ℤ_≥‘𝑀))
65		eluzfz1 13263	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
66	64, 65	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
67		rspcsbela 4369	. . . . . . . . 9 ⊢ ((𝑀 ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
68	66, 67	sylancom 588	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
69		eluzfz2 13264	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘𝑀) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1)))
70	64, 69	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1)))
71		rspcsbela 4369	. . . . . . . . 9 ⊢ (((𝑀 + 1) ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
72	70, 71	sylancom 588	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
73		telgsumfzs.m	. . . . . . . . 9 ⊢ − = (-_g‘𝐺)
74	54, 73	grpsubcl 18655	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
75	60, 68, 72, 74	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
76		csbeq1 3835	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → ⦋𝑖 / 𝑘⦌𝐶 = ⦋𝑀 / 𝑘⦌𝐶)
77		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1))
78	77	csbeq1d 3836	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶)
79	76, 78	oveq12d 7293	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
80	79	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) ∧ 𝑖 = 𝑀) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
81	54, 59, 49, 75, 80	gsumsnd 19553	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ {𝑀} ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
82	53, 81	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑀) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
83	54, 55, 73	telgsumfzslem 19589	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ((𝐺 Σ_g (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶) → (𝐺 Σ_g (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)))
84	83	ex 413	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ((𝐺 Σ_g (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶) → (𝐺 Σ_g (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))))
85		eluzelz 12592	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → 𝑦 ∈ ℤ)
86	85	peano2zd 12429	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ ℤ)
87	86	peano2zd 12429	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ ℤ)
88		peano2z 12361	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
89	88	zred 12426	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℝ)
90	85, 89	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ ℝ)
91	90	lep1d 11906	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1))
92		eluz2 12588	. . . . . . . . . 10 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)) ↔ ((𝑦 + 1) ∈ ℤ ∧ ((𝑦 + 1) + 1) ∈ ℤ ∧ (𝑦 + 1) ≤ ((𝑦 + 1) + 1)))
93	86, 87, 91, 92	syl3anbrc 1342	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
94		fzss2 13296	. . . . . . . . 9 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)))
95	93, 94	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)))
96		ssralv 3987	. . . . . . . 8 ⊢ ((𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
97	95, 96	syl 17	. . . . . . 7 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
98	97	adantld 491	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
99	84, 98	a2and 842	. . . . 5 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶)) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))))
100	13, 24, 35, 46, 82, 99	uzind4i 12650	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵) → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶)))
101	100	expd 416	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵 → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶))))
102	2, 101	mpcom 38	. 2 ⊢ (𝜑 → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵 → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶)))
103	1, 102	mpd 15	1 ⊢ (𝜑 → (𝐺 Σ_g (𝑖 ∈ (𝑀...𝑁) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⦋csb 3832 ⊆ wss 3887 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 1c1 10872 + caddc 10874 ≤ cle 11010 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 Basecbs 16912 Σ_g cgsu 17151 Mndcmnd 18385 Grpcgrp 18577 -_gcsg 18579 Abelcabl 19387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-0g 17152 df-gsum 17153 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-abl 19389

This theorem is referenced by: telgsumfz 19591 telgsumfz0s 19592

W3C validator