MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  telgsumfzs Structured version   Visualization version   GIF version

Theorem telgsumfzs 19571
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 7275 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑥 + 1) = (𝑀 + 1))
43oveq2d 7284 . . . . . . . 8 (𝑥 = 𝑀 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑀 + 1)))
54raleqdv 3346 . . . . . . 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 7276 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑀...𝑥) = (𝑀...𝑀))
87mpteq1d 5173 . . . . . . . 8 (𝑥 = 𝑀 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
98oveq2d 7284 . . . . . . 7 (𝑥 = 𝑀 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
103csbeq1d 3840 . . . . . . . 8 (𝑥 = 𝑀(𝑥 + 1) / 𝑘𝐶 = (𝑀 + 1) / 𝑘𝐶)
1110oveq2d 7284 . . . . . . 7 (𝑥 = 𝑀 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
129, 11eqeq12d 2755 . . . . . 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 7275 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 + 1) = (𝑦 + 1))
1514oveq2d 7284 . . . . . . . 8 (𝑥 = 𝑦 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑦 + 1)))
1615raleqdv 3346 . . . . . . 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 7276 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑀...𝑥) = (𝑀...𝑦))
1918mpteq1d 5173 . . . . . . . 8 (𝑥 = 𝑦 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
2019oveq2d 7284 . . . . . . 7 (𝑥 = 𝑦 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑦) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
2114csbeq1d 3840 . . . . . . . 8 (𝑥 = 𝑦(𝑥 + 1) / 𝑘𝐶 = (𝑦 + 1) / 𝑘𝐶)
2221oveq2d 7284 . . . . . . 7 (𝑥 = 𝑦 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑦 + 1) / 𝑘𝐶))
2320, 22eqeq12d 2755 . . . . . 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 7275 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑥 + 1) = ((𝑦 + 1) + 1))
2625oveq2d 7284 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑀...(𝑥 + 1)) = (𝑀...((𝑦 + 1) + 1)))
2726raleqdv 3346 . . . . . . 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 7276 . . . . . . . . 9 (𝑥 = (𝑦 + 1) → (𝑀...𝑥) = (𝑀...(𝑦 + 1)))
3029mpteq1d 5173 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
3130oveq2d 7284 . . . . . . 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 3840 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝑥 + 1) / 𝑘𝐶 = ((𝑦 + 1) + 1) / 𝑘𝐶)
3332oveq2d 7284 . . . . . . 7 (𝑥 = (𝑦 + 1) → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 ((𝑦 + 1) + 1) / 𝑘𝐶))
3431, 33eqeq12d 2755 . . . . . 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 7275 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1))
3736oveq2d 7284 . . . . . . . 8 (𝑥 = 𝑁 → (𝑀...(𝑥 + 1)) = (𝑀...(𝑁 + 1)))
3837raleqdv 3346 . . . . . . 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 7276 . . . . . . . . 9 (𝑥 = 𝑁 → (𝑀...𝑥) = (𝑀...𝑁))
4140mpteq1d 5173 . . . . . . . 8 (𝑥 = 𝑁 → (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
4241oveq2d 7284 . . . . . . 7 (𝑥 = 𝑁 → (𝐺 Σg (𝑖 ∈ (𝑀...𝑥) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝐺 Σg (𝑖 ∈ (𝑀...𝑁) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))))
4336csbeq1d 3840 . . . . . . . 8 (𝑥 = 𝑁(𝑥 + 1) / 𝑘𝐶 = (𝑁 + 1) / 𝑘𝐶)
4443oveq2d 7284 . . . . . . 7 (𝑥 = 𝑁 → (𝑀 / 𝑘𝐶 (𝑥 + 1) / 𝑘𝐶) = (𝑀 / 𝑘𝐶 (𝑁 + 1) / 𝑘𝐶))
4542, 44eqeq12d 2755 . . . . . 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 12569 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
482, 47syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
4948adantr 480 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ ℤ)
50 fzsn 13280 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
5149, 50syl 17 . . . . . . . 8 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀...𝑀) = {𝑀})
5251mpteq1d 5173 . . . . . . 7 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑖 ∈ (𝑀...𝑀) ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)) = (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶)))
5352oveq2d 7284 . . . . . 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 19372 . . . . . . . . . 10 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
5755, 56syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
5857grpmndd 18570 . . . . . . . 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 12579 . . . . . . . . . . . 12 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
6249, 61syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ (ℤ𝑀))
63 peano2uz 12623 . . . . . . . . . . 11 (𝑀 ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (ℤ𝑀))
6462, 63syl 17 . . . . . . . . . 10 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 + 1) ∈ (ℤ𝑀))
65 eluzfz1 13245 . . . . . . . . . 10 ((𝑀 + 1) ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
6664, 65syl 17 . . . . . . . . 9 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → 𝑀 ∈ (𝑀...(𝑀 + 1)))
67 rspcsbela 4374 . . . . . . . . 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 13246 . . . . . . . . . 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 4374 . . . . . . . . 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 18636 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑀 / 𝑘𝐶𝐵(𝑀 + 1) / 𝑘𝐶𝐵) → (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶) ∈ 𝐵)
7560, 68, 72, 74syl3anc 1369 . . . . . . 7 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶) ∈ 𝐵)
76 csbeq1 3839 . . . . . . . . 9 (𝑖 = 𝑀𝑖 / 𝑘𝐶 = 𝑀 / 𝑘𝐶)
77 oveq1 7275 . . . . . . . . . 10 (𝑖 = 𝑀 → (𝑖 + 1) = (𝑀 + 1))
7877csbeq1d 3840 . . . . . . . . 9 (𝑖 = 𝑀(𝑖 + 1) / 𝑘𝐶 = (𝑀 + 1) / 𝑘𝐶)
7976, 78oveq12d 7286 . . . . . . . 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 19534 . . . . . 6 ((𝜑 ∧ ∀𝑘 ∈ (𝑀...(𝑀 + 1))𝐶𝐵) → (𝐺 Σg (𝑖 ∈ {𝑀} ↦ (𝑖 / 𝑘𝐶 (𝑖 + 1) / 𝑘𝐶))) = (𝑀 / 𝑘𝐶 (𝑀 + 1) / 𝑘𝐶))
8253, 81eqtrd 2779 . . . . 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 19570 . . . . . . 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 12574 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → 𝑦 ∈ ℤ)
8685peano2zd 12411 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ∈ ℤ)
8786peano2zd 12411 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → ((𝑦 + 1) + 1) ∈ ℤ)
88 peano2z 12344 . . . . . . . . . . . . 13 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
8988zred 12408 . . . . . . . . . . . 12 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℝ)
9085, 89syl 17 . . . . . . . . . . 11 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ∈ ℝ)
9190lep1d 11889 . . . . . . . . . 10 (𝑦 ∈ (ℤ𝑀) → (𝑦 + 1) ≤ ((𝑦 + 1) + 1))
92 eluz2 12570 . . . . . . . . . 10 (((𝑦 + 1) + 1) ∈ (ℤ‘(𝑦 + 1)) ↔ ((𝑦 + 1) ∈ ℤ ∧ ((𝑦 + 1) + 1) ∈ ℤ ∧ (𝑦 + 1) ≤ ((𝑦 + 1) + 1)))
9386, 87, 91, 92syl3anbrc 1341 . . . . . . . . 9 (𝑦 ∈ (ℤ𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
94 fzss2 13278 . . . . . . . . 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 3991 . . . . . . . 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 841 . . . . 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 12632 . . . 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 1541  wcel 2109  wral 3065  csb 3836  wss 3891  {csn 4566   class class class wbr 5078  cmpt 5161  cfv 6430  (class class class)co 7268  cr 10854  1c1 10856   + caddc 10858  cle 10994  cz 12302  cuz 12564  ...cfz 13221  Basecbs 16893   Σg cgsu 17132  Mndcmnd 18366  Grpcgrp 18558  -gcsg 18560  Abelcabl 19368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524  df-om 7701  df-1st 7817  df-2nd 7818  df-supp 7962  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-fsupp 9090  df-oi 9230  df-card 9681  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-2 12019  df-n0 12217  df-z 12303  df-uz 12565  df-fz 13222  df-fzo 13365  df-seq 13703  df-hash 14026  df-sets 16846  df-slot 16864  df-ndx 16876  df-base 16894  df-ress 16923  df-plusg 16956  df-0g 17133  df-gsum 17134  df-mre 17276  df-mrc 17277  df-acs 17279  df-mgm 18307  df-sgrp 18356  df-mnd 18367  df-submnd 18412  df-grp 18561  df-minusg 18562  df-sbg 18563  df-mulg 18682  df-cntz 18904  df-cmn 19369  df-abl 19370
This theorem is referenced by:  telgsumfz  19572  telgsumfz0s  19573
  Copyright terms: Public domain W3C validator