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Theorem telgsumfzslem 19589

Description: Lemma for telgsumfzs 19590 (induction step). (Contributed by AV, 23-Nov-2019.)

**Hypotheses**
Ref	Expression
telgsumfzs.b	⊢ 𝐵 = (Base‘𝐺)
telgsumfzs.g	⊢ (𝜑 → 𝐺 ∈ Abel)
telgsumfzs.m	⊢ − = (-_g‘𝐺)

**Assertion**
Ref	Expression
telgsumfzslem	⊢ ((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)))

Distinct variable groups: 𝐵,𝑖,𝑘 𝐶,𝑖 𝑖,𝐺 𝑖,𝑀,𝑘 − ,𝑖 𝜑,𝑖 𝑦,𝑖,𝑘 Allowed substitution hints: 𝜑(𝑦,𝑘) 𝐵(𝑦) 𝐶(𝑦,𝑘) 𝐺(𝑦,𝑘) 𝑀(𝑦) − (𝑦,𝑘)

**Proof of Theorem telgsumfzslem**
Step	Hyp	Ref	Expression
1		telgsumfzs.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2738	. . . . 5 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
3		telgsumfzs.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Abel)
4	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Abel)
5		ablcmn 19393	. . . . . . 7 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ CMnd)
7	6	adantl 482	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ CMnd)
8		fzfid 13693	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑀...(𝑦 + 1)) ∈ Fin)
9		ablgrp 19391	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
10	3, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp)
11	10	ad2antrl 725	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Grp)
12	11	adantr 481	. . . . . 6 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → 𝐺 ∈ Grp)
13		fzelp1 13308	. . . . . . 7 ⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → 𝑖 ∈ (𝑀...((𝑦 + 1) + 1)))
14		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)
15	14	adantl 482	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)
16		rspcsbela 4369	. . . . . . 7 ⊢ ((𝑖 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵)
17	13, 15, 16	syl2anr 597	. . . . . 6 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵)
18		fzp1elp1 13309	. . . . . . 7 ⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → (𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
19		rspcsbela 4369	. . . . . . 7 ⊢ (((𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
20	18, 15, 19	syl2anr 597	. . . . . 6 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
21		telgsumfzs.m	. . . . . . 7 ⊢ − = (-_g‘𝐺)
22	1, 21	grpsubcl 18655	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
23	12, 17, 20, 22	syl3anc 1370	. . . . 5 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) ∈ 𝐵)
24		fzp1disj 13315	. . . . . 6 ⊢ ((𝑀...𝑦) ∩ {(𝑦 + 1)}) = ∅
25	24	a1i 11	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ((𝑀...𝑦) ∩ {(𝑦 + 1)}) = ∅)
26		fzsuc 13303	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
27	26	adantr 481	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
28	1, 2, 7, 8, 23, 25, 27	gsummptfidmsplit 19531	. . . 4 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝐺 Σ_g (𝑖 ∈ (𝑀...(𝑦 + 1)) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = ((𝐺 Σ_g (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))(+_g‘𝐺)(𝐺 Σ_g (𝑖 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))))
29	28	adantr 481	. . 3 ⊢ (((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))) = ((𝐺 Σ_g (𝑖 ∈ (𝑀...𝑦) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))(+_g‘𝐺)(𝐺 Σ_g (𝑖 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))))
30		simpr 485	. . . 4 ⊢ (((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) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶))
31	10	grpmndd 18589	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
32	31	ad2antrl 725	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Mnd)
33		ovexd 7310	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑦 + 1) ∈ V)
34		peano2uz 12641	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (ℤ_≥‘𝑀))
35		eluzfz2 13264	. . . . . . . . . 10 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...(𝑦 + 1)))
36	34, 35	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...(𝑦 + 1)))
37		fzelp1 13308	. . . . . . . . 9 ⊢ ((𝑦 + 1) ∈ (𝑀...(𝑦 + 1)) → (𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
38	36, 37	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
39		rspcsbela 4369	. . . . . . . 8 ⊢ (((𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
40	38, 14, 39	syl2an 596	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
41		peano2uz 12641	. . . . . . . . . 10 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘𝑀))
42	34, 41	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘𝑀))
43		eluzfz2 13264	. . . . . . . . 9 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
44	42, 43	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
45		rspcsbela 4369	. . . . . . . 8 ⊢ ((((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵)
46	44, 14, 45	syl2an 596	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵)
47	1, 21	grpsubcl 18655	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵) → (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) ∈ 𝐵)
48	11, 40, 46, 47	syl3anc 1370	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) ∈ 𝐵)
49		csbeq1 3835	. . . . . . . 8 ⊢ (𝑖 = (𝑦 + 1) → ⦋𝑖 / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶)
50		oveq1 7282	. . . . . . . . 9 ⊢ (𝑖 = (𝑦 + 1) → (𝑖 + 1) = ((𝑦 + 1) + 1))
51	50	csbeq1d 3836	. . . . . . . 8 ⊢ (𝑖 = (𝑦 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)
52	49, 51	oveq12d 7293	. . . . . . 7 ⊢ (𝑖 = (𝑦 + 1) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
53	52	adantl 482	. . . . . 6 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 = (𝑦 + 1)) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
54	1, 32, 33, 48, 53	gsumsnd 19553	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝐺 Σ_g (𝑖 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
55	54	adantr 481	. . . 4 ⊢ (((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 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶))) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
56	30, 55	oveq12d 7293	. . 3 ⊢ (((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) ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))(+_g‘𝐺)(𝐺 Σ_g (𝑖 ∈ {(𝑦 + 1)} ↦ (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶)))) = ((⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶)(+_g‘𝐺)(⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)))
57		eluzfz1 13263	. . . . . . 7 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...((𝑦 + 1) + 1)))
58	42, 57	syl 17	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...((𝑦 + 1) + 1)))
59		rspcsbela 4369	. . . . . 6 ⊢ ((𝑀 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
60	58, 14, 59	syl2an 596	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
61	1, 2, 21	grpnpncan 18670	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵 ∧ ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵)) → ((⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶)(+_g‘𝐺)(⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
62	11, 60, 40, 46, 61	syl13anc 1371	. . . 4 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ((⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶)(+_g‘𝐺)(⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
63	62	adantr 481	. . 3 ⊢ (((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)) → ((⦋𝑀 / 𝑘⦌𝐶 − ⦋(𝑦 + 1) / 𝑘⦌𝐶)(+_g‘𝐺)(⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)) = (⦋𝑀 / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
64	29, 56, 63	3eqtrd 2782	. 2 ⊢ (((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))
65	64	ex 413	1 ⊢ ((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)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⦋csb 3832 ∪ cun 3885 ∩ cin 3886 ∅c0 4256 {csn 4561 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 1c1 10872 + caddc 10874 ℤ_≥cuz 12582 ...cfz 13239 Basecbs 16912 +_gcplusg 16962 Σ_g cgsu 17151 Mndcmnd 18385 Grpcgrp 18577 -_gcsg 18579 CMndccmn 19386 Abelcabl 19387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-0g 17152 df-gsum 17153 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-abl 19389

This theorem is referenced by: telgsumfzs 19590

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