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Theorem telgsumfzs 20065

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 7419	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1))
4	3	oveq2d 7428	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1)))
5	4	raleqdv 3322	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵))
6	5	anbi2d 641	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵)))
7		oveq2 7420	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
8	7	mpteq1d 5200	. . . . . . . 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 7428	. . . . . . 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 3856	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶)
11	10	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
12	9, 11	eqeq12d 2778	. . . . . 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 7419	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
15	14	oveq2d 7428	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1)))
16	15	raleqdv 3322	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
17	16	anbi2d 641	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵)))
18		oveq2 7420	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦))
19	18	mpteq1d 5200	. . . . . . . 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 7428	. . . . . . 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 3856	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶)
22	21	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶))
23	20, 22	eqeq12d 2778	. . . . . 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 7419	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
26	25	oveq2d 7428	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1)))
27	26	raleqdv 3322	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵))
28	27	anbi2d 641	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)))
29		oveq2 7420	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1)))
30	29	mpteq1d 5200	. . . . . . . 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 7428	. . . . . . 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 3856	. . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)
33	32	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
34	31, 33	eqeq12d 2778	. . . . . 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 7419	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
37	36	oveq2d 7428	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1)))
38	37	raleqdv 3322	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵))
39	38	anbi2d 641	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶 ∈ 𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶 ∈ 𝐵)))
40		oveq2 7420	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
41	40	mpteq1d 5200	. . . . . . . 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 7428	. . . . . . 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 3856	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ⦋(𝑥 + 1) / 𝑘⦌𝐶 = ⦋(𝑁 + 1) / 𝑘⦌𝐶)
44	43	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑥 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑁 + 1) / 𝑘⦌𝐶))
45	42, 44	eqeq12d 2778	. . . . . 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 12873	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
48	2, 47	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
49	48	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ ℤ)
50		fzsn 13601	. . . . . . . . 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 5200	. . . . . . 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 7428	. . . . . 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 19861	. . . . . . . . . 10 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
57	55, 56	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Grp)
58	57	grpmndd 19019	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
59	58	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Mnd)
60	57	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Grp)
61		uzid 12883	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
62	49, 61	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → 𝑀 ∈ (ℤ_≥‘𝑀))
63		peano2uz 12931	. . . . . . . . . . 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 13565	. . . . . . . . . 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 4402	. . . . . . . . 9 ⊢ ((𝑀 ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
68	66, 67	sylancom 599	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
69		eluzfz2 13566	. . . . . . . . . 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 4402	. . . . . . . . 9 ⊢ (((𝑀 + 1) ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
72	70, 71	sylancom 599	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → ⦋(𝑀 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
73		telgsumfzs.m	. . . . . . . . 9 ⊢ − = (-_g‘𝐺)
74	54, 73	grpsubcl 19092	. . . . . . . 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 1397	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶 ∈ 𝐵) → (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
76		csbeq1 3855	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → ⦋𝑖 / 𝑘⦌𝐶 = ⦋𝑀 / 𝑘⦌𝐶)
77		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1))
78	77	csbeq1d 3856	. . . . . . . . 9 ⊢ (𝑖 = 𝑀 → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋(𝑀 + 1) / 𝑘⦌𝐶)
79	76, 78	oveq12d 7430	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑀 + 1) / 𝑘⦌𝐶))
80	79	adantl 486	. . . . . . 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 20028	. . . . . 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 2797	. . . . 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 20064	. . . . . . 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 417	. . . . . 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 12878	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → 𝑦 ∈ ℤ)
86	85	peano2zd 12709	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ ℤ)
87	86	peano2zd 12709	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ ℤ)
88		peano2z 12641	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
89	88	zred 12706	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℝ)
90	85, 89	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ ℝ)
91	90	lep1d 12152	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1))
92		eluz2 12874	. . . . . . . . . 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 1361	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
94		fzss2 13599	. . . . . . . . 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 4005	. . . . . . . 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 495	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶 ∈ 𝐵))
99	84, 98	a2and 858	. . . . 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 12940	. . . 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 420	. . 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 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 ⦋csb 3852 ⊆ wss 3904 {csn 4588 class class class wbr 5108 ↦ cmpt 5191 ‘cfv 6536 (class class class)co 7412 ℝcr 11105 1c1 11107 + caddc 11109 ≤ cle 11250 ℤcz 12597 ℤ_≥cuz 12868 ...cfz 13541 Basecbs 17275 Σ_g cgsu 17499 Mndcmnd 18798 Grpcgrp 19006 -_gcsg 19008 Abelcabl 19857

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-n0 12511 df-z 12598 df-uz 12869 df-fz 13542 df-fzo 13690 df-seq 14045 df-hash 14374 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-0g 17500 df-gsum 17501 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-abl 19859

This theorem is used by: telgsumfz 20066 telgsumfz0s 20067

W3C validator