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Theorem telgsumfzslem 20116

Description: Lemma for telgsumfzs 20117 (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 2762	. . . . 5 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
3		telgsumfzs.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Abel)
4	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ Abel)
5		ablcmn 19915	. . . . . . 7 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
6	4, 5	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → 𝐺 ∈ CMnd)
7	6	adantl 487	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ CMnd)
8		fzfid 14039	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑀...(𝑦 + 1)) ∈ Fin)
9		ablgrp 19913	. . . . . . . . 9 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
10	3, 9	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp)
11	10	ad2antrl 741	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Grp)
12	11	adantr 486	. . . . . 6 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → 𝐺 ∈ Grp)
13		fzelp1 13633	. . . . . . 7 ⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → 𝑖 ∈ (𝑀...((𝑦 + 1) + 1)))
14		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)
15	14	adantl 487	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)
16		rspcsbela 4399	. . . . . . 7 ⊢ ((𝑖 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵)
17	13, 15, 16	syl2anr 609	. . . . . 6 ⊢ (((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) ∧ 𝑖 ∈ (𝑀...(𝑦 + 1))) → ⦋𝑖 / 𝑘⦌𝐶 ∈ 𝐵)
18		fzp1elp1 13634	. . . . . . 7 ⊢ (𝑖 ∈ (𝑀...(𝑦 + 1)) → (𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
19		rspcsbela 4399	. . . . . . 7 ⊢ (((𝑖 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
20	18, 15, 19	syl2anr 609	. . . . . 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 19144	. . . . . 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 1398	. . . . 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 13640	. . . . . 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 13628	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
27	26	adantr 486	. . . . 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 20058	. . . 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 486	. . 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 490	. . . 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 19071	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mnd)
32	31	ad2antrl 741	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → 𝐺 ∈ Mnd)
33		ovexd 7451	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (𝑦 + 1) ∈ V)
34		peano2uz 12953	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (ℤ_≥‘𝑀))
35		eluzfz2 13588	. . . . . . . . . 10 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...(𝑦 + 1)))
36	34, 35	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...(𝑦 + 1)))
37		fzelp1 13633	. . . . . . . . 9 ⊢ ((𝑦 + 1) ∈ (𝑀...(𝑦 + 1)) → (𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
38	36, 37	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → (𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
39		rspcsbela 4399	. . . . . . . 8 ⊢ (((𝑦 + 1) ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
40	38, 14, 39	syl2an 608	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋(𝑦 + 1) / 𝑘⦌𝐶 ∈ 𝐵)
41		peano2uz 12953	. . . . . . . . . 10 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘𝑀))
42	34, 41	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘𝑀))
43		eluzfz2 13588	. . . . . . . . 9 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
44	42, 43	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → ((𝑦 + 1) + 1) ∈ (𝑀...((𝑦 + 1) + 1)))
45		rspcsbela 4399	. . . . . . . 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 608	. . . . . . 7 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶 ∈ 𝐵)
47	1, 21	grpsubcl 19144	. . . . . . 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 1398	. . . . . 6 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶) ∈ 𝐵)
49		csbeq1 3853	. . . . . . . 8 ⊢ (𝑖 = (𝑦 + 1) → ⦋𝑖 / 𝑘⦌𝐶 = ⦋(𝑦 + 1) / 𝑘⦌𝐶)
50		oveq1 7423	. . . . . . . . 9 ⊢ (𝑖 = (𝑦 + 1) → (𝑖 + 1) = ((𝑦 + 1) + 1))
51	50	csbeq1d 3854	. . . . . . . 8 ⊢ (𝑖 = (𝑦 + 1) → ⦋(𝑖 + 1) / 𝑘⦌𝐶 = ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶)
52	49, 51	oveq12d 7434	. . . . . . 7 ⊢ (𝑖 = (𝑦 + 1) → (⦋𝑖 / 𝑘⦌𝐶 − ⦋(𝑖 + 1) / 𝑘⦌𝐶) = (⦋(𝑦 + 1) / 𝑘⦌𝐶 − ⦋((𝑦 + 1) + 1) / 𝑘⦌𝐶))
53	52	adantl 487	. . . . . 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 20080	. . . . 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 486	. . . 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 7434	. . 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 13587	. . . . . . 7 ⊢ (((𝑦 + 1) + 1) ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...((𝑦 + 1) + 1)))
58	42, 57	syl 18	. . . . . 6 ⊢ (𝑦 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...((𝑦 + 1) + 1)))
59		rspcsbela 4399	. . . . . 6 ⊢ ((𝑀 ∈ (𝑀...((𝑦 + 1) + 1)) ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
60	58, 14, 59	syl2an 608	. . . . 5 ⊢ ((𝑦 ∈ (ℤ_≥‘𝑀) ∧ (𝜑 ∧ ∀𝑘 ∈ (𝑀...((𝑦 + 1) + 1))𝐶 ∈ 𝐵)) → ⦋𝑀 / 𝑘⦌𝐶 ∈ 𝐵)
61	1, 2, 21	grpnpncan 19159	. . . . 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 1399	. . . 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 486	. . 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 2801	. 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 418	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3078 Vcvv 3453 ⦋csb 3850 ∪ cun 3900 ∩ cin 3901 ∅c0 4282 {csn 4587 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7416 1c1 11128 + caddc 11130 ℤ_≥cuz 12890 ...cfz 13563 Basecbs 17305 +_gcplusg 17346 Σ_g cgsu 17529 Mndcmnd 18838 Grpcgrp 19058 -_gcsg 19060 CMndccmn 19908 Abelcabl 19909

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13564 df-fzo 13712 df-seq 14068 df-hash 14397 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-0g 17530 df-gsum 17531 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-abl 19911

This theorem is used by: telgsumfzs 20117

W3C validator