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Theorem telgsumfzs 18594
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 6800 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1))
43oveq2d 6809 . . . . . . . 8 (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1)))
54raleqdv 3293 . . . . . . 7 (𝑥 = 𝑀 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵))
65anbi2d 614 . . . . . 6 (𝑥 = 𝑀 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵)))
7 oveq2 6801 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
87mpteq1d 4872 . . . . . . . 8 (𝑥 = 𝑀 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
98oveq2d 6809 . . . . . . 7 (𝑥 = 𝑀 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
103csbeq1d 3689 . . . . . . . 8 (𝑥 = 𝑀(𝑥 + 1) / 𝑘𝐶 = (𝑀 + 1) / 𝑘𝐶)
1110oveq2d 6809 . . . . . . 7 (𝑥 = 𝑀 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
129, 11eqeq12d 2786 . . . . . 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 333 . . . . 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 6800 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
1514oveq2d 6809 . . . . . . . 8 (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1)))
1615raleqdv 3293 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
1716anbi2d 614 . . . . . 6 (𝑥 = 𝑦 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵)))
18 oveq2 6801 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦))
1918mpteq1d 4872 . . . . . . . 8 (𝑥 = 𝑦 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
2019oveq2d 6809 . . . . . . 7 (𝑥 = 𝑦 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
2114csbeq1d 3689 . . . . . . . 8 (𝑥 = 𝑦(𝑥 + 1) / 𝑘𝐶 = (𝑦 + 1) / 𝑘𝐶)
2221oveq2d 6809 . . . . . . 7 (𝑥 = 𝑦 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶))
2320, 22eqeq12d 2786 . . . . . 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 333 . . . . 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 6800 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
2625oveq2d 6809 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1)))
2726raleqdv 3293 . . . . . . 7 (𝑥 = (𝑦 + 1) → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵 ↔ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵))
2827anbi2d 614 . . . . . 6 (𝑥 = (𝑦 + 1) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵)))
29 oveq2 6801 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1)))
3029mpteq1d 4872 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
3130oveq2d 6809 . . . . . . 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 3689 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 + 1) / 𝑘𝐶 = ((𝑦 + 1) + 1) / 𝑘𝐶)
3332oveq2d 6809 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶))
3431, 33eqeq12d 2786 . . . . . 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 333 . . . . 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 6800 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
3736oveq2d 6809 . . . . . . . 8 (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1)))
3837raleqdv 3293 . . . . . . 7 (𝑥 = 𝑁 → (∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵 ↔ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵))
3938anbi2d 614 . . . . . 6 (𝑥 = 𝑁 → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑥 + 1))𝐶𝐵) ↔ (𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵)))
40 oveq2 6801 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
4140mpteq1d 4872 . . . . . . . 8 (𝑥 = 𝑁 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
4241oveq2d 6809 . . . . . . 7 (𝑥 = 𝑁 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
4336csbeq1d 3689 . . . . . . . 8 (𝑥 = 𝑁(𝑥 + 1) / 𝑘𝐶 = (𝑁 + 1) / 𝑘𝐶)
4443oveq2d 6809 . . . . . . 7 (𝑥 = 𝑁 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))
4542, 44eqeq12d 2786 . . . . . 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 333 . . . . 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 11893 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
482, 47syl 17 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
4948adantr 466 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ ℤ)
50 fzsn 12590 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
5149, 50syl 17 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀...𝑀) = {𝑀})
5251mpteq1d 4872 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
5352oveq2d 6809 . . . . . . 7 ((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 . . . . . . . 8 𝐵 = (Base‘𝐺)
55 telgsumfzs.g . . . . . . . . . . 11 (𝜑𝐺 ∈ Abel)
56 ablgrp 18405 . . . . . . . . . . 11 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
5755, 56syl 17 . . . . . . . . . 10 (𝜑𝐺 ∈ Grp)
58 grpmnd 17637 . . . . . . . . . 10 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
5957, 58syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Mnd)
6059adantr 466 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝐺 ∈ Mnd)
6157adantr 466 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝐺 ∈ Grp)
62 uzid 11903 . . . . . . . . . . . . 13 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
6349, 62syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ (ℤ𝑀))
64 peano2uz 11943 . . . . . . . . . . . 12 (𝑀 ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (ℤ𝑀))
6563, 64syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) ∈ (ℤ𝑀))
66 eluzfz1 12555 . . . . . . . . . . 11 ((𝑀 + 1) ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
6765, 66syl 17 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
68 rspcsbela 4150 . . . . . . . . . 10 ((𝑀 ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 / 𝑘𝐶𝐵)
6967, 68sylancom 576 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 / 𝑘𝐶𝐵)
70 eluzfz2 12556 . . . . . . . . . . 11 ((𝑀 + 1) ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1)))
7165, 70syl 17 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) ∈ (𝑀...(𝑀 + 1)))
72 rspcsbela 4150 . . . . . . . . . 10 (((𝑀 + 1) ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) / 𝑘𝐶𝐵)
7371, 72sylancom 576 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) / 𝑘𝐶𝐵)
74 telgsumfzs.m . . . . . . . . . 10 = (-g𝐺)
7554, 74grpsubcl 17703 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑀 / 𝑘𝐶𝐵(𝑀 + 1) / 𝑘𝐶𝐵) → (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶) ∈ 𝐵)
7661, 69, 73, 75syl3anc 1476 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶) ∈ 𝐵)
77 csbeq1 3685 . . . . . . . . . 10 (𝑖 = 𝑀𝑖 / 𝑘𝐶 = 𝑀 / 𝑘𝐶)
78 oveq1 6800 . . . . . . . . . . 11 (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1))
7978csbeq1d 3689 . . . . . . . . . 10 (𝑖 = 𝑀(𝑖 + 1) / 𝑘𝐶 = (𝑀 + 1) / 𝑘𝐶)
8077, 79oveq12d 6811 . . . . . . . . 9 (𝑖 = 𝑀 → (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8180adantl 467 . . . . . . . 8 (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) ∧ 𝑖 = 𝑀) → (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8254, 60, 49, 76, 81gsumsnd 18559 . . . . . . 7 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8353, 82eqtrd 2805 . . . . . 6 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8483a1i 11 . . . . 5 (𝑀 ∈ ℤ → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶)))
8554, 55, 74telgsumfzslem 18593 . . . . . . 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)))
8685ex 397 . . . . . 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))))
87 eluzelz 11898 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → 𝑦 ∈ ℤ)
8887peano2zd 11687 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ∈ ℤ)
8988peano2zd 11687 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → ((𝑦 + 1) + 1) ∈ ℤ)
90 peano2z 11620 . . . . . . . . . . . . 13 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
9190zred 11684 . . . . . . . . . . . 12 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℝ)
9287, 91syl 17 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ∈ ℝ)
9392lep1d 11157 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1))
94 eluz2 11894 . . . . . . . . . 10 (((𝑦 + 1) + 1) ∈ (ℤ‘(𝑦 + 1)) ↔ ((𝑦 + 1) ∈ ℤ ∧ ((𝑦 + 1) + 1) ∈ ℤ ∧ (𝑦 + 1) ≤ ((𝑦 + 1) + 1)))
9588, 89, 93, 94syl3anbrc 1428 . . . . . . . . 9 (𝑦 ∈ (ℤ𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
96 fzss2 12588 . . . . . . . . 9 (((𝑦 + 1) + 1) ∈ (ℤ‘(𝑦 + 1)) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)))
9795, 96syl 17 . . . . . . . 8 (𝑦 ∈ (ℤ𝑀) → (𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)))
98 ssralv 3815 . . . . . . . 8 ((𝑀...(𝑦 + 1)) ⊆ (𝑀...((𝑦 + 1) + 1)) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
9997, 98syl 17 . . . . . . 7 (𝑦 ∈ (ℤ𝑀) → (∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵 → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
10099adantld 478 . . . . . 6 (𝑦 ∈ (ℤ𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
10186, 100a2and 834 . . . . 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))))
10213, 24, 35, 46, 84, 101uzind4 11948 . . . 4 (𝑁 ∈ (ℤ𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶)))
103102expd 400 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))))
1042, 103mpcom 38 . 2 (𝜑 → (∀𝑘 ∈ (𝑀...(𝑁 + 1))𝐶𝐵 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶)))
1051, 104mpd 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 382   = wceq 1631  wcel 2145  wral 3061  csb 3682  wss 3723  {csn 4316   class class class wbr 4786  cmpt 4863  cfv 6031  (class class class)co 6793  cr 10137  1c1 10139   + caddc 10141  cle 10277  cz 11579  cuz 11888  ...cfz 12533  Basecbs 16064   Σg cgsu 16309  Mndcmnd 17502  Grpcgrp 17630  -gcsg 17632  Abelcabl 18401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16310  df-gsum 16311  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-mulg 17749  df-cntz 17957  df-cmn 18402  df-abl 18403
This theorem is referenced by:  telgsumfz  18595  telgsumfz0s  18596
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