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Theorem telgsumfzs 20164

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 7415	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1))
4	3	oveq2d 7424	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1)))
5	4	raleqdv 3319	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵))
6	5	anbi2d 642	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵)))
7		oveq2 7416	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
8	7	mpteq1d 5194	. . . . . . . 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 7424	. . . . . . 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 3850	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶)
11	10	oveq2d 7424	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
12	9, 11	eqeq12d 2776	. . . . . 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 347	. . . . 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 7415	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
15	14	oveq2d 7424	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1)))
16	15	raleqdv 3319	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
17	16	anbi2d 642	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)))
18		oveq2 7416	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦))
19	18	mpteq1d 5194	. . . . . . . 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 7424	. . . . . . 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 3850	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶)
22	21	oveq2d 7424	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶))
23	20, 22	eqeq12d 2776	. . . . . 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 347	. . . . 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 7415	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
26	25	oveq2d 7424	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1)))
27	26	raleqdv 3319	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵))
28	27	anbi2d 642	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)))
29		oveq2 7416	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1)))
30	29	mpteq1d 5194	. . . . . . . 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 7424	. . . . . . 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 3850	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)
33	32	oveq2d 7424	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
34	31, 33	eqeq12d 2776	. . . . . 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 347	. . . . 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 7415	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
37	36	oveq2d 7424	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1)))
38	37	raleqdv 3319	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵))
39	38	anbi2d 642	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵)))
40		oveq2 7416	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
41	40	mpteq1d 5194	. . . . . . . 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 7424	. . . . . . 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 3850	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑁 + 1) / 𝑘⦌𝐶)
44	43	oveq2d 7424	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶))
45	42, 44	eqeq12d 2776	. . . . . 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 347	. . . . 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 12939	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
48	2, 47	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
49	48	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ ℤ)
50		fzsn 13668	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
51	49, 50	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀...𝑀) = {𝑀})
52	51	mpteq1d 5194	. . . . . . 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 7424	. . . . . 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 19960	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
57	55, 56	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp)
58	57	grpmndd 19118	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
59	58	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Mnd)
60	57	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Grp)
61		uzid 12949	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
62	49, 61	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (ℤ_≥‘𝑀))
63		peano2uz 12997	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘𝑀) → (𝑀 + 1) ∈ (ℤ_≥‘𝑀))
64	62, 63	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀 + 1) ∈ (ℤ_≥‘𝑀))
65		eluzfz1 13632	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
66	64, 65	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
67		rspcsbela 4395	. . . . . . . . 9 ⊢ ((𝑀 ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
68	66, 67	sylancom 600	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
69		eluzfz2 13633	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘𝑀) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1)))
70	64, 69	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1)))
71		rspcsbela 4395	. . . . . . . . 9 ⊢ (((𝑀 + 1) ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
72	70, 71	sylancom 600	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
73		telgsumfzs.m	. . . . . . . . 9 ⊢ − = (-_g‘𝐺)
74	54, 73	grpsubcl 19191	. . . . . . . 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 1398	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
76		csbeq1 3849	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → ⦋𝑖 / 𝑘⦌𝐶 = ⦋𝑀 / 𝑘⦌𝐶)
77		oveq1 7415	. . . . . . . . . 10 ⊢ (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1))
78	77	csbeq1d 3850	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶)
79	76, 78	oveq12d 7426	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
80	79	adantl 487	. . . . . . 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 20127	. . . . . 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 2795	. . . . 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 20163	. . . . . . 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 418	. . . . . 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 12944	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → 𝑦 ∈ ℤ)
86	85	peano2zd 12775	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ ℤ)
87	86	peano2zd 12775	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ ℤ)
88		peano2z 12706	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
89	88	zred 12772	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℝ)
90	85, 89	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ ℝ)
91	90	lep1d 12217	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1))
92		eluz2 12940	. . . . . . . . . 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 1362	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
94		fzss2 13666	. . . . . . . . 9 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)))
95	93, 94	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)))
96		ssralv 3999	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
98	97	adantld 496	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
99	84, 98	a2and 859	. . . . 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 13006	. . . 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 421	. . 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 39	. 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 16	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ⦋csb 3846 ⊆ wss 3898 {csn 4583 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6527 (class class class)co 7408 ℝcr 11170 1c1 11172 + caddc 11174 ≤ cle 11315 ℤcz 12662 ℤ_≥cuz 12934 ...cfz 13608 Basecbs 17348 Σ_g cgsu 17572 Mndcmnd 18884 Grpcgrp 19105 -_gcsg 19107 Abelcabl 19956

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-er 8695 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-oi 9482 df-card 9991 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-nn 12305 df-2 12374 df-n0 12576 df-z 12663 df-uz 12935 df-fz 13609 df-fzo 13757 df-seq 14113 df-hash 14442 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-ress 17370 df-plusg 17402 df-0g 17573 df-gsum 17574 df-mre 17717 df-mrc 17718 df-acs 17720 df-mgm 18777 df-sgrp 18869 df-mnd 18885 df-submnd 18940 df-grp 19108 df-minusg 19109 df-sbg 19110 df-mulg 19239 df-cntz 19492 df-cmn 19957 df-abl 19958

This theorem is used by: telgsumfz 20165 telgsumfz0s 20166

W3C validator