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Theorem cvg1nlemcau 11006

Description: Lemma for cvg1n 11008. By selecting spaced out terms for the modified sequence 𝐺, the terms are within 1 / 𝑛 (without the constant 𝐶). (Contributed by Jim Kingdon, 1-Aug-2021.)

**Hypotheses**
Ref	Expression
cvg1n.f	⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
cvg1n.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
cvg1n.cau	⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))
cvg1nlem.g	⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))
cvg1nlem.z	⊢ (𝜑 → 𝑍 ∈ ℕ)
cvg1nlem.start	⊢ (𝜑 → 𝐶 < 𝑍)

**Assertion**
Ref	Expression
cvg1nlemcau	⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛))))

Distinct variable groups: 𝐶,𝑛,𝑘 𝑛,𝐹,𝑗,𝑘 𝑗,𝑍 𝜑,𝑘,𝑛 Allowed substitution hints: 𝜑(𝑗) 𝐶(𝑗) 𝐺(𝑗,𝑘,𝑛) 𝑍(𝑘,𝑛)

Proof of Theorem cvg1nlemcau Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ)
2		cvg1n.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
3	2	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐹:ℕ⟶ℝ)
4		cvg1nlem.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ ℕ)
5	4	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑍 ∈ ℕ)
6	1, 5	nnmulcld 8981	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 · 𝑍) ∈ ℕ)
7	3, 6	ffvelcdmd 5665	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘(𝑛 · 𝑍)) ∈ ℝ)
8		oveq1 5895	. . . . . . . . 9 ⊢ (𝑗 = 𝑛 → (𝑗 · 𝑍) = (𝑛 · 𝑍))
9	8	fveq2d 5531	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (𝐹‘(𝑗 · 𝑍)) = (𝐹‘(𝑛 · 𝑍)))
10		cvg1nlem.g	. . . . . . . 8 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))
11	9, 10	fvmptg 5605	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐹‘(𝑛 · 𝑍)) ∈ ℝ) → (𝐺‘𝑛) = (𝐹‘(𝑛 · 𝑍)))
12	1, 7, 11	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐺‘𝑛) = (𝐹‘(𝑛 · 𝑍)))
13	12, 7	eqeltrd 2264	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐺‘𝑛) ∈ ℝ)
14		eluznn 9613	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
15	14	adantll 476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
16	15, 5	nnmulcld 8981	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑘 · 𝑍) ∈ ℕ)
17	3, 16	ffvelcdmd 5665	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘(𝑘 · 𝑍)) ∈ ℝ)
18		oveq1 5895	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 · 𝑍) = (𝑘 · 𝑍))
19	18	fveq2d 5531	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐹‘(𝑗 · 𝑍)) = (𝐹‘(𝑘 · 𝑍)))
20	19, 10	fvmptg 5605	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝐹‘(𝑘 · 𝑍)) ∈ ℝ) → (𝐺‘𝑘) = (𝐹‘(𝑘 · 𝑍)))
21	15, 17, 20	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐺‘𝑘) = (𝐹‘(𝑘 · 𝑍)))
22	21, 17	eqeltrd 2264	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐺‘𝑘) ∈ ℝ)
23		cvg1n.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
24	23	rpred 9709	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
25	24	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐶 ∈ ℝ)
26	25, 6	nndivred 8982	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐶 / (𝑛 · 𝑍)) ∈ ℝ)
27	22, 26	readdcld 8000	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ∈ ℝ)
28	1	nnrecred 8979	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (1 / 𝑛) ∈ ℝ)
29	22, 28	readdcld 8000	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑘) + (1 / 𝑛)) ∈ ℝ)
30		fveq2 5527	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑘 · 𝑍) → (𝐹‘𝑏) = (𝐹‘(𝑘 · 𝑍)))
31	30	oveq1d 5903	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑘 · 𝑍) → ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))) = ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))
32	31	breq2d 4027	. . . . . . . . . 10 ⊢ (𝑏 = (𝑘 · 𝑍) → ((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))) ↔ (𝐹‘(𝑛 · 𝑍)) < ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
33	30	breq1d 4025	. . . . . . . . . 10 ⊢ (𝑏 = (𝑘 · 𝑍) → ((𝐹‘𝑏) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))) ↔ (𝐹‘(𝑘 · 𝑍)) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
34	32, 33	anbi12d 473	. . . . . . . . 9 ⊢ (𝑏 = (𝑘 · 𝑍) → (((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐹‘𝑏) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))) ↔ ((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐹‘(𝑘 · 𝑍)) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))))
35		fveq2 5527	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑛 · 𝑍) → (ℤ_≥‘𝑎) = (ℤ_≥‘(𝑛 · 𝑍)))
36		fveq2 5527	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑛 · 𝑍) → (𝐹‘𝑎) = (𝐹‘(𝑛 · 𝑍)))
37		oveq2 5896	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑛 · 𝑍) → (𝐶 / 𝑎) = (𝐶 / (𝑛 · 𝑍)))
38	37	oveq2d 5904	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑛 · 𝑍) → ((𝐹‘𝑏) + (𝐶 / 𝑎)) = ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))))
39	36, 38	breq12d 4028	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑛 · 𝑍) → ((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ↔ (𝐹‘(𝑛 · 𝑍)) < ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍)))))
40	36, 37	oveq12d 5906	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑛 · 𝑍) → ((𝐹‘𝑎) + (𝐶 / 𝑎)) = ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))
41	40	breq2d 4027	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑛 · 𝑍) → ((𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎)) ↔ (𝐹‘𝑏) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
42	39, 41	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑛 · 𝑍) → (((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎))) ↔ ((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐹‘𝑏) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))))
43	35, 42	raleqbidv 2695	. . . . . . . . . 10 ⊢ (𝑎 = (𝑛 · 𝑍) → (∀𝑏 ∈ (ℤ_≥‘𝑎)((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎))) ↔ ∀𝑏 ∈ (ℤ_≥‘(𝑛 · 𝑍))((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐹‘𝑏) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))))
44		cvg1n.cau	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))
45		fveq2 5527	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
46	45	oveq1d 5903	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) + (𝐶 / 𝑛)) = ((𝐹‘𝑏) + (𝐶 / 𝑛)))
47	46	breq2d 4027	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ↔ (𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛))))
48	45	breq1d 4025	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛)) ↔ (𝐹‘𝑏) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))
49	47, 48	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑏 → (((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))) ↔ ((𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑛) + (𝐶 / 𝑛)))))
50	49	cbvralv 2715	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))) ↔ ∀𝑏 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))
51	50	ralbii 2493	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))) ↔ ∀𝑛 ∈ ℕ ∀𝑏 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑛) + (𝐶 / 𝑛))))
52		fveq2 5527	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑎))
53		fveq2 5527	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → (𝐹‘𝑛) = (𝐹‘𝑎))
54		oveq2 5896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑎 → (𝐶 / 𝑛) = (𝐶 / 𝑎))
55	54	oveq2d 5904	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑏) + (𝐶 / 𝑛)) = ((𝐹‘𝑏) + (𝐶 / 𝑎)))
56	53, 55	breq12d 4028	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛)) ↔ (𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎))))
57	53, 54	oveq12d 5906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) + (𝐶 / 𝑛)) = ((𝐹‘𝑎) + (𝐶 / 𝑎)))
58	57	breq2d 4027	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑏) < ((𝐹‘𝑛) + (𝐶 / 𝑛)) ↔ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎))))
59	56, 58	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → (((𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑛) + (𝐶 / 𝑛))) ↔ ((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎)))))
60	52, 59	raleqbidv 2695	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑎 → (∀𝑏 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑛) + (𝐶 / 𝑛))) ↔ ∀𝑏 ∈ (ℤ_≥‘𝑎)((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎)))))
61	60	cbvralv 2715	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ ℕ ∀𝑏 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑛) + (𝐶 / 𝑛))) ↔ ∀𝑎 ∈ ℕ ∀𝑏 ∈ (ℤ_≥‘𝑎)((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎))))
62	51, 61	bitri 184	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (𝐶 / 𝑛))) ↔ ∀𝑎 ∈ ℕ ∀𝑏 ∈ (ℤ_≥‘𝑎)((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎))))
63	44, 62	sylib 122	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑎 ∈ ℕ ∀𝑏 ∈ (ℤ_≥‘𝑎)((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎))))
64	63	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ∀𝑎 ∈ ℕ ∀𝑏 ∈ (ℤ_≥‘𝑎)((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ∧ (𝐹‘𝑏) < ((𝐹‘𝑎) + (𝐶 / 𝑎))))
65	43, 64, 6	rspcdva 2858	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ∀𝑏 ∈ (ℤ_≥‘(𝑛 · 𝑍))((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐹‘𝑏) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
66		eluzle 9553	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → 𝑛 ≤ 𝑘)
67	66	adantl 277	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ≤ 𝑘)
68	1	nnred 8945	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℝ)
69	15	nnred 8945	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℝ)
70	5	nnrpd 9707	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑍 ∈ ℝ⁺)
71	68, 69, 70	lemul1d 9753	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 ≤ 𝑘 ↔ (𝑛 · 𝑍) ≤ (𝑘 · 𝑍)))
72	67, 71	mpbid 147	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 · 𝑍) ≤ (𝑘 · 𝑍))
73	6	nnzd 9387	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 · 𝑍) ∈ ℤ)
74	16	nnzd 9387	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑘 · 𝑍) ∈ ℤ)
75		eluz 9554	. . . . . . . . . . 11 ⊢ (((𝑛 · 𝑍) ∈ ℤ ∧ (𝑘 · 𝑍) ∈ ℤ) → ((𝑘 · 𝑍) ∈ (ℤ_≥‘(𝑛 · 𝑍)) ↔ (𝑛 · 𝑍) ≤ (𝑘 · 𝑍)))
76	73, 74, 75	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝑘 · 𝑍) ∈ (ℤ_≥‘(𝑛 · 𝑍)) ↔ (𝑛 · 𝑍) ≤ (𝑘 · 𝑍)))
77	72, 76	mpbird 167	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑘 · 𝑍) ∈ (ℤ_≥‘(𝑛 · 𝑍)))
78	34, 65, 77	rspcdva 2858	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐹‘(𝑘 · 𝑍)) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
79	21	oveq1d 5903	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) = ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))
80	79	breq2d 4027	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘(𝑛 · 𝑍)) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ↔ (𝐹‘(𝑛 · 𝑍)) < ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
81	21	breq1d 4025	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑘) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))) ↔ (𝐹‘(𝑘 · 𝑍)) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
82	80, 81	anbi12d 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((𝐹‘(𝑛 · 𝑍)) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐺‘𝑘) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))) ↔ ((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐹‘(𝑘 · 𝑍)) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))))
83	78, 82	mpbird 167	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘(𝑛 · 𝑍)) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐺‘𝑘) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
84	12	breq1d 4025	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑛) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ↔ (𝐹‘(𝑛 · 𝑍)) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍)))))
85	12	oveq1d 5903	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍))) = ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))
86	85	breq2d 4027	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑘) < ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍))) ↔ (𝐺‘𝑘) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
87	84, 86	anbi12d 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((𝐺‘𝑛) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍)))) ↔ ((𝐹‘(𝑛 · 𝑍)) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐺‘𝑘) < ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))))
88	83, 87	mpbird 167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑛) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍)))))
89	88	simpld 112	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐺‘𝑛) < ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))))
90	5	nnred 8945	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑍 ∈ ℝ)
91	1	nnrpd 9707	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℝ⁺)
92		cvg1nlem.start	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 < 𝑍)
93	92	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐶 < 𝑍)
94	25, 90, 91, 93	ltmul1dd 9765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐶 · 𝑛) < (𝑍 · 𝑛))
95	6	nncnd 8946	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 · 𝑍) ∈ ℂ)
96	95	mulid2d 7989	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (1 · (𝑛 · 𝑍)) = (𝑛 · 𝑍))
97	96	breq2d 4027	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐶 · 𝑛) < (1 · (𝑛 · 𝑍)) ↔ (𝐶 · 𝑛) < (𝑛 · 𝑍)))
98		1red 7985	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 1 ∈ ℝ)
99	6	nnrpd 9707	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 · 𝑍) ∈ ℝ⁺)
100	25, 91, 98, 99	lt2mul2divd 9778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐶 · 𝑛) < (1 · (𝑛 · 𝑍)) ↔ (𝐶 / (𝑛 · 𝑍)) < (1 / 𝑛)))
101	1	nncnd 8946	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℂ)
102	5	nncnd 8946	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑍 ∈ ℂ)
103	101, 102	mulcomd 7992	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 · 𝑍) = (𝑍 · 𝑛))
104	103	breq2d 4027	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐶 · 𝑛) < (𝑛 · 𝑍) ↔ (𝐶 · 𝑛) < (𝑍 · 𝑛)))
105	97, 100, 104	3bitr3d 218	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐶 / (𝑛 · 𝑍)) < (1 / 𝑛) ↔ (𝐶 · 𝑛) < (𝑍 · 𝑛)))
106	94, 105	mpbird 167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐶 / (𝑛 · 𝑍)) < (1 / 𝑛))
107	26, 28, 22, 106	ltadd2dd 8392	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) < ((𝐺‘𝑘) + (1 / 𝑛)))
108	13, 27, 29, 89, 107	lttrd 8096	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)))
109	13, 26	readdcld 8000	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍))) ∈ ℝ)
110	13, 28	readdcld 8000	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑛) + (1 / 𝑛)) ∈ ℝ)
111	88	simprd 114	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐺‘𝑘) < ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍))))
112	26, 28, 13, 106	ltadd2dd 8392	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍))) < ((𝐺‘𝑛) + (1 / 𝑛)))
113	22, 109, 110, 111, 112	lttrd 8096	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛)))
114	108, 113	jca 306	. . 3 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛))))
115	114	ralrimiva 2560	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛))))
116	115	ralrimiva 2560	1 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 ∀wral 2465 class class class wbr 4015 ↦ cmpt 4076 ⟶wf 5224 ‘cfv 5228 (class class class)co 5888 ℝcr 7823 1c1 7825 + caddc 7827 · cmul 7829 < clt 8005 ≤ cle 8006 / cdiv 8642 ℕcn 8932 ℤcz 9266 ℤ_≥cuz 9541 ℝ⁺crp 9666

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942

This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-n0 9190 df-z 9267 df-uz 9542 df-rp 9667

This theorem is referenced by: cvg1nlemres 11007

W3C validator