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Theorem telgsumfzs 19909
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 (𝜑𝐺 ∈ Abel)
telgsumfzs.m = (-g𝐺)
telgsumfzs.n (𝜑𝑁 ∈ (ℤ𝑀))
telgsumfzs.f (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵)
Assertion
Ref Expression
telgsumfzs (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))
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.
StepHypRef Expression
1 telgsumfzs.f . 2 (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵)
2 telgsumfzs.n . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
3 oveq1 7412 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1))
43oveq2d 7421 . . . . . . . 8 (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1)))
54raleqdv 3319 . . . . . . 7 (𝑥 = 𝑀 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵))
65anbi2d 628 . . . . . 6 (𝑥 = 𝑀 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵)))
7 oveq2 7413 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
87mpteq1d 5236 . . . . . . . 8 (𝑥 = 𝑀 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
98oveq2d 7421 . . . . . . 7 (𝑥 = 𝑀 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
103csbeq1d 3892 . . . . . . . 8 (𝑥 = 𝑀(𝑥 + 1) / 𝑘𝐶 = (𝑀 + 1) / 𝑘𝐶)
1110oveq2d 7421 . . . . . . 7 (𝑥 = 𝑀 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
129, 11eqeq12d 2742 . . . . . 6 (𝑥 = 𝑀 → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) ↔ (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶)))
136, 12imbi12d 344 . . . . 5 (𝑥 = 𝑀 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))))
14 oveq1 7412 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
1514oveq2d 7421 . . . . . . . 8 (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1)))
1615raleqdv 3319 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
1716anbi2d 628 . . . . . 6 (𝑥 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵)))
18 oveq2 7413 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦))
1918mpteq1d 5236 . . . . . . . 8 (𝑥 = 𝑦 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
2019oveq2d 7421 . . . . . . 7 (𝑥 = 𝑦 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
2114csbeq1d 3892 . . . . . . . 8 (𝑥 = 𝑦(𝑥 + 1) / 𝑘𝐶 = (𝑦 + 1) / 𝑘𝐶)
2221oveq2d 7421 . . . . . . 7 (𝑥 = 𝑦 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶))
2320, 22eqeq12d 2742 . . . . . 6 (𝑥 = 𝑦 → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) ↔ (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶)))
2417, 23imbi12d 344 . . . . 5 (𝑥 = 𝑦 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶))))
25 oveq1 7412 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
2625oveq2d 7421 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1)))
2726raleqdv 3319 . . . . . . 7 (𝑥 = (𝑦 + 1) → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵 ↔ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵))
2827anbi2d 628 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵)))
29 oveq2 7413 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1)))
3029mpteq1d 5236 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
3130oveq2d 7421 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
3225csbeq1d 3892 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 + 1) / 𝑘𝐶 = ((𝑦 + 1) + 1) / 𝑘𝐶)
3332oveq2d 7421 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶))
3431, 33eqeq12d 2742 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) ↔ (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶)))
3528, 34imbi12d 344 . . . . 5 (𝑥 = (𝑦 + 1) → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶))))
36 oveq1 7412 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
3736oveq2d 7421 . . . . . . . 8 (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1)))
3837raleqdv 3319 . . . . . . 7 (𝑥 = 𝑁 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵))
3938anbi2d 628 . . . . . 6 (𝑥 = 𝑁 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵)))
40 oveq2 7413 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
4140mpteq1d 5236 . . . . . . . 8 (𝑥 = 𝑁 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
4241oveq2d 7421 . . . . . . 7 (𝑥 = 𝑁 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
4336csbeq1d 3892 . . . . . . . 8 (𝑥 = 𝑁(𝑥 + 1) / 𝑘𝐶 = (𝑁 + 1) / 𝑘𝐶)
4443oveq2d 7421 . . . . . . 7 (𝑥 = 𝑁 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))
4542, 44eqeq12d 2742 . . . . . 6 (𝑥 = 𝑁 → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) ↔ (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶)))
4639, 45imbi12d 344 . . . . 5 (𝑥 = 𝑁 → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶)) ↔ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))))
47 eluzel2 12831 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
482, 47syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
4948adantr 480 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ ℤ)
50 fzsn 13549 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
5149, 50syl 17 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀...𝑀) = {𝑀})
5251mpteq1d 5236 . . . . . . 7 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
5352oveq2d 7421 . . . . . 6 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
54 telgsumfzs.b . . . . . . 7 𝐵 = (Base‘𝐺)
55 telgsumfzs.g . . . . . . . . . 10 (𝜑𝐺 ∈ Abel)
56 ablgrp 19705 . . . . . . . . . 10 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
5755, 56syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
5857grpmndd 18876 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
5958adantr 480 . . . . . . 7 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝐺 ∈ Mnd)
6057adantr 480 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝐺 ∈ Grp)
61 uzid 12841 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
6249, 61syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ (ℤ𝑀))
63 peano2uz 12889 . . . . . . . . . . 11 (𝑀 ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (ℤ𝑀))
6462, 63syl 17 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) ∈ (ℤ𝑀))
65 eluzfz1 13514 . . . . . . . . . 10 ((𝑀 + 1) ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
6664, 65syl 17 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
67 rspcsbela 4430 . . . . . . . . 9 ((𝑀 ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 / 𝑘𝐶𝐵)
6866, 67sylancom 587 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 / 𝑘𝐶𝐵)
69 eluzfz2 13515 . . . . . . . . . 10 ((𝑀 + 1) ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1)))
7064, 69syl 17 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1)))
71 rspcsbela 4430 . . . . . . . . 9 (((𝑀 + 1) ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) / 𝑘𝐶𝐵)
7270, 71sylancom 587 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) / 𝑘𝐶𝐵)
73 telgsumfzs.m . . . . . . . . 9 = (-g𝐺)
7454, 73grpsubcl 18948 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑀 / 𝑘𝐶𝐵(𝑀 + 1) / 𝑘𝐶𝐵) → (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶) ∈ 𝐵)
7560, 68, 72, 74syl3anc 1368 . . . . . . 7 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶) ∈ 𝐵)
76 csbeq1 3891 . . . . . . . . 9 (𝑖 = 𝑀𝑖 / 𝑘𝐶 = 𝑀 / 𝑘𝐶)
77 oveq1 7412 . . . . . . . . . 10 (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1))
7877csbeq1d 3892 . . . . . . . . 9 (𝑖 = 𝑀(𝑖 + 1) / 𝑘𝐶 = (𝑀 + 1) / 𝑘𝐶)
7976, 78oveq12d 7423 . . . . . . . 8 (𝑖 = 𝑀 → (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8079adantl 481 . . . . . . 7 (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) ∧ 𝑖 = 𝑀) → (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8154, 59, 49, 75, 80gsumsnd 19872 . . . . . 6 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8253, 81eqtrd 2766 . . . . 5 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8354, 55, 73telgsumfzslem 19908 . . . . . . 7 ((𝑦 ∈ (ℤ𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵)) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶)))
8483ex 412 . . . . . 6 (𝑦 ∈ (ℤ𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵) → ((𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶))))
85 eluzelz 12836 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → 𝑦 ∈ ℤ)
8685peano2zd 12673 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ∈ ℤ)
8786peano2zd 12673 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → ((𝑦 + 1) + 1) ∈ ℤ)
88 peano2z 12607 . . . . . . . . . . . . 13 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
8988zred 12670 . . . . . . . . . . . 12 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℝ)
9085, 89syl 17 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ∈ ℝ)
9190lep1d 12149 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1))
92 eluz2 12832 . . . . . . . . . 10 (((𝑦 + 1) + 1) ∈ (ℤ‘(𝑦 + 1)) ↔ ((𝑦 + 1) ∈ ℤ ∧ ((𝑦 + 1) + 1) ∈ ℤ ∧ (𝑦 + 1) ≤ ((𝑦 + 1) + 1)))
9386, 87, 91, 92syl3anbrc 1340 . . . . . . . . 9 (𝑦 ∈ (ℤ𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
94 fzss2 13547 . . . . . . . . 9 (((𝑦 + 1) + 1) ∈ (ℤ‘(𝑦 + 1)) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)))
9593, 94syl 17 . . . . . . . 8 (𝑦 ∈ (ℤ𝑀) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)))
96 ssralv 4045 . . . . . . . 8 ((𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
9795, 96syl 17 . . . . . . 7 (𝑦 ∈ (ℤ𝑀) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
9897adantld 490 . . . . . 6 (𝑦 ∈ (ℤ𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
9984, 98a2and 842 . . . . 5 (𝑦 ∈ (ℤ𝑀) → (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶)) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶))))
10013, 24, 35, 46, 82, 99uzind4i 12898 . . . 4 (𝑁 ∈ (ℤ𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶)))
101100expd 415 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))))
1022, 101mpcom 38 . 2 (𝜑 → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶)))
1031, 102mpd 15 1 (𝜑 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  csb 3888  wss 3943  {csn 4623   class class class wbr 5141  cmpt 5224  cfv 6537  (class class class)co 7405  cr 11111  1c1 11113   + caddc 11115  cle 11253  cz 12562  cuz 12826  ...cfz 13490  Basecbs 17153   Σg cgsu 17395  Mndcmnd 18667  Grpcgrp 18863  -gcsg 18865  Abelcabl 19701
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14296  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-gsum 17397  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18996  df-cntz 19233  df-cmn 19702  df-abl 19703
This theorem is referenced by:  telgsumfz  19910  telgsumfz0s  19911
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