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Theorem telgsumfzs 19961
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 7426 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1))
43oveq2d 7435 . . . . . . . 8 (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1)))
54raleqdv 3314 . . . . . . 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 7427 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
87mpteq1d 5244 . . . . . . . 8 (𝑥 = 𝑀 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
98oveq2d 7435 . . . . . . 7 (𝑥 = 𝑀 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
103csbeq1d 3893 . . . . . . . 8 (𝑥 = 𝑀(𝑥 + 1) / 𝑘𝐶 = (𝑀 + 1) / 𝑘𝐶)
1110oveq2d 7435 . . . . . . 7 (𝑥 = 𝑀 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
129, 11eqeq12d 2741 . . . . . 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 343 . . . . 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 7426 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
1514oveq2d 7435 . . . . . . . 8 (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1)))
1615raleqdv 3314 . . . . . . 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 7427 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦))
1918mpteq1d 5244 . . . . . . . 8 (𝑥 = 𝑦 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
2019oveq2d 7435 . . . . . . 7 (𝑥 = 𝑦 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
2114csbeq1d 3893 . . . . . . . 8 (𝑥 = 𝑦(𝑥 + 1) / 𝑘𝐶 = (𝑦 + 1) / 𝑘𝐶)
2221oveq2d 7435 . . . . . . 7 (𝑥 = 𝑦 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶))
2320, 22eqeq12d 2741 . . . . . 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 343 . . . . 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 7426 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
2625oveq2d 7435 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1)))
2726raleqdv 3314 . . . . . . 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 7427 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1)))
3029mpteq1d 5244 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
3130oveq2d 7435 . . . . . . 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 3893 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 + 1) / 𝑘𝐶 = ((𝑦 + 1) + 1) / 𝑘𝐶)
3332oveq2d 7435 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶))
3431, 33eqeq12d 2741 . . . . . 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 343 . . . . 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 7426 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
3736oveq2d 7435 . . . . . . . 8 (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1)))
3837raleqdv 3314 . . . . . . 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 7427 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
4140mpteq1d 5244 . . . . . . . 8 (𝑥 = 𝑁 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
4241oveq2d 7435 . . . . . . 7 (𝑥 = 𝑁 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
4336csbeq1d 3893 . . . . . . . 8 (𝑥 = 𝑁(𝑥 + 1) / 𝑘𝐶 = (𝑁 + 1) / 𝑘𝐶)
4443oveq2d 7435 . . . . . . 7 (𝑥 = 𝑁 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))
4542, 44eqeq12d 2741 . . . . . 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 343 . . . . 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 12865 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
482, 47syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
4948adantr 479 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ ℤ)
50 fzsn 13583 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
5149, 50syl 17 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀...𝑀) = {𝑀})
5251mpteq1d 5244 . . . . . . 7 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
5352oveq2d 7435 . . . . . 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 19757 . . . . . . . . . 10 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
5755, 56syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
5857grpmndd 18916 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
5958adantr 479 . . . . . . 7 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝐺 ∈ Mnd)
6057adantr 479 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝐺 ∈ Grp)
61 uzid 12875 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
6249, 61syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ (ℤ𝑀))
63 peano2uz 12923 . . . . . . . . . . 11 (𝑀 ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (ℤ𝑀))
6462, 63syl 17 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) ∈ (ℤ𝑀))
65 eluzfz1 13548 . . . . . . . . . 10 ((𝑀 + 1) ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
6664, 65syl 17 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
67 rspcsbela 4437 . . . . . . . . 9 ((𝑀 ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 / 𝑘𝐶𝐵)
6866, 67sylancom 586 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 / 𝑘𝐶𝐵)
69 eluzfz2 13549 . . . . . . . . . 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 4437 . . . . . . . . 9 (((𝑀 + 1) ∈ (𝑀...(𝑀 + 1)) ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) / 𝑘𝐶𝐵)
7270, 71sylancom 586 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) / 𝑘𝐶𝐵)
73 telgsumfzs.m . . . . . . . . 9 = (-g𝐺)
7454, 73grpsubcl 18989 . . . . . . . 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 3892 . . . . . . . . 9 (𝑖 = 𝑀𝑖 / 𝑘𝐶 = 𝑀 / 𝑘𝐶)
77 oveq1 7426 . . . . . . . . . 10 (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1))
7877csbeq1d 3893 . . . . . . . . 9 (𝑖 = 𝑀(𝑖 + 1) / 𝑘𝐶 = (𝑀 + 1) / 𝑘𝐶)
7976, 78oveq12d 7437 . . . . . . . 8 (𝑖 = 𝑀 → (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8079adantl 480 . . . . . . 7 (((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) ∧ 𝑖 = 𝑀) → (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8154, 59, 49, 75, 80gsumsnd 19924 . . . . . 6 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8253, 81eqtrd 2765 . . . . 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 19960 . . . . . . 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 411 . . . . . 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 12870 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → 𝑦 ∈ ℤ)
8685peano2zd 12707 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ∈ ℤ)
8786peano2zd 12707 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → ((𝑦 + 1) + 1) ∈ ℤ)
88 peano2z 12641 . . . . . . . . . . . . 13 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
8988zred 12704 . . . . . . . . . . . 12 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℝ)
9085, 89syl 17 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ∈ ℝ)
9190lep1d 12183 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1))
92 eluz2 12866 . . . . . . . . . 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 13581 . . . . . . . . 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 489 . . . . . 6 (𝑦 ∈ (ℤ𝑀) → ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶𝐵) → ∀𝑘 ∈ (𝑀...(𝑦 + 1))𝐶𝐵))
9984, 98a2and 843 . . . . 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 12932 . . . 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 414 . . 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 394   = wceq 1533  wcel 2098  wral 3050  csb 3889  wss 3944  {csn 4630   class class class wbr 5149  cmpt 5232  cfv 6549  (class class class)co 7419  cr 11144  1c1 11146   + caddc 11148  cle 11286  cz 12596  cuz 12860  ...cfz 13524  Basecbs 17188   Σg cgsu 17430  Mndcmnd 18702  Grpcgrp 18903  -gcsg 18905  Abelcabl 19753
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9393  df-oi 9540  df-card 9969  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-seq 14008  df-hash 14331  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17189  df-ress 17218  df-plusg 17254  df-0g 17431  df-gsum 17432  df-mre 17574  df-mrc 17575  df-acs 17577  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18749  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19037  df-cntz 19285  df-cmn 19754  df-abl 19755
This theorem is referenced by:  telgsumfz  19962  telgsumfz0s  19963
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