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Theorem telgsumfzslem 20030

Description: Lemma for telgsumfzs 20031 (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 2764	. . . . 5 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
3		telgsumfzs.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Abel)
4	3	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Abel)
5		ablcmn 19829	. . . . . . 7 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ CMnd)
7	6	adantl 485	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ CMnd)
8		fzfid 13988	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑀...(𝑦 + 1)) ∈ Fin)
9		ablgrp 19827	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
10	3, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp)
11	10	ad2antrl 738	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Grp)
12	11	adantr 484	. . . . . 6 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → 𝐺 ∈ Grp)
13		fzelp1 13583	. . . . . . 7 ⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → 𝑖 ∈ (𝑀...((𝑦 + 1) + 1)))
14		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)
15	14	adantl 485	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)
16		rspcsbela 4394	. . . . . . 7 ⊢ ((𝑖 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵)
17	13, 15, 16	syl2anr 606	. . . . . 6 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵)
18		fzp1elp1 13584	. . . . . . 7 ⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → (𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
19		rspcsbela 4394	. . . . . . 7 ⊢ (((𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
20	18, 15, 19	syl2anr 606	. . . . . 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 19064	. . . . . 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 1392	. . . . 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 13590	. . . . . 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 13578	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
27	26	adantr 484	. . . . 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 19972	. . . 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 484	. . 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 488	. . . 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 18990	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
32	31	ad2antrl 738	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Mnd)
33		ovexd 7433	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑦 + 1) ∈ V)
34		peano2uz 12904	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (ℤ_≥‘𝑀))
35		eluzfz2 13539	. . . . . . . . . 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 13583	. . . . . . . . 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 4394	. . . . . . . 8 ⊢ (((𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
40	38, 14, 39	syl2an 605	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
41		peano2uz 12904	. . . . . . . . . 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 13539	. . . . . . . . 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 4394	. . . . . . . 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 605	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵)
47	1, 21	grpsubcl 19064	. . . . . . 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 1392	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) ∈ 𝐵)
49		csbeq1 3857	. . . . . . . 8 ⊢ (𝑖 = (𝑦 + 1) → ⦋𝑖 / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶)
50		oveq1 7405	. . . . . . . . 9 ⊢ (𝑖 = (𝑦 + 1) → (𝑖 + 1) = ((𝑦 + 1) + 1))
51	50	csbeq1d 3858	. . . . . . . 8 ⊢ (𝑖 = (𝑦 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)
52	49, 51	oveq12d 7416	. . . . . . 7 ⊢ (𝑖 = (𝑦 + 1) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
53	52	adantl 485	. . . . . 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 19994	. . . . 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 484	. . . 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 7416	. . 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 13538	. . . . . . 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 4394	. . . . . 6 ⊢ ((𝑀 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
60	58, 14, 59	syl2an 605	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
61	1, 2, 21	grpnpncan 19079	. . . . 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 1393	. . . 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 484	. . 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 2803	. 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 416	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 399 = wceq 1562 ∈ wcel 2144 ∀wral 3078 Vcvv 3456 ⦋csb 3854 ∪ cun 3904 ∩ cin 3905 ∅c0 4287 {csn 4584 ↦ cmpt 5183 ‘cfv 6523 (class class class)co 7398 1c1 11076 + caddc 11078 ℤ_≥cuz 12841 ...cfz 13514 Basecbs 17247 +_gcplusg 17288 Σ_g cgsu 17471 Mndcmnd 18770 Grpcgrp 18977 -_gcsg 18979 CMndccmn 19822 Abelcabl 19823

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-nn 12213 df-2 12282 df-n0 12484 df-z 12571 df-uz 12842 df-fz 13515 df-fzo 13662 df-seq 14017 df-hash 14346 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-0g 17472 df-gsum 17473 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-grp 18980 df-minusg 18981 df-sbg 18982 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-abl 19825

This theorem is referenced by: telgsumfzs 20031

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