Theorem cvg1nlemcau 11538

Description: Lemma for cvg1n 11540. 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 9185	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝑛 · 𝑍) ∈ ℕ)
7	3, 6	ffvelcdmd 5779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘(𝑛 · 𝑍)) ∈ ℝ)
8		oveq1 6020	. . . . . . . . 9 ⊢ (𝑗 = 𝑛 → (𝑗 · 𝑍) = (𝑛 · 𝑍))
9	8	fveq2d 5639	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (𝐹‘(𝑗 · 𝑍)) = (𝐹‘(𝑛 · 𝑍)))
10		cvg1nlem.g	. . . . . . . 8 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ (𝐹‘(𝑗 · 𝑍)))
11	9, 10	fvmptg 5718	. . . . . . 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 2306	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐺‘𝑛) ∈ ℝ)
14		eluznn 9827	. . . . . . . . 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 9185	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝑘 · 𝑍) ∈ ℕ)
17	3, 16	ffvelcdmd 5779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘(𝑘 · 𝑍)) ∈ ℝ)
18		oveq1 6020	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑗 · 𝑍) = (𝑘 · 𝑍))
19	18	fveq2d 5639	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐹‘(𝑗 · 𝑍)) = (𝐹‘(𝑘 · 𝑍)))
20	19, 10	fvmptg 5718	. . . . . . . 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 2306	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐺‘𝑘) ∈ ℝ)
23		cvg1n.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
24	23	rpred 9924	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
25	24	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝐶 ∈ ℝ)
26	25, 6	nndivred 9186	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐶 / (𝑛 · 𝑍)) ∈ ℝ)
27	22, 26	readdcld 8202	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) ∈ ℝ)
28	1	nnrecred 9183	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1 / 𝑛) ∈ ℝ)
29	22, 28	readdcld 8202	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐺‘𝑘) + (1 / 𝑛)) ∈ ℝ)
30		fveq2 5635	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑘 · 𝑍) → (𝐹‘𝑏) = (𝐹‘(𝑘 · 𝑍)))
31	30	oveq1d 6028	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑘 · 𝑍) → ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))) = ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))
32	31	breq2d 4098	. . . . . . . . . 10 ⊢ (𝑏 = (𝑘 · 𝑍) → ((𝐹‘(𝑛 · 𝑍)) < ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))) ↔ (𝐹‘(𝑛 · 𝑍)) < ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍)))))
33	30	breq1d 4096	. . . . . . . . . 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 5635	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑛 · 𝑍) → (ℤ≥‘𝑎) = (ℤ≥‘(𝑛 · 𝑍)))
36		fveq2 5635	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑛 · 𝑍) → (𝐹‘𝑎) = (𝐹‘(𝑛 · 𝑍)))
37		oveq2 6021	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑛 · 𝑍) → (𝐶 / 𝑎) = (𝐶 / (𝑛 · 𝑍)))
38	37	oveq2d 6029	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑛 · 𝑍) → ((𝐹‘𝑏) + (𝐶 / 𝑎)) = ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍))))
39	36, 38	breq12d 4099	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑛 · 𝑍) → ((𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎)) ↔ (𝐹‘(𝑛 · 𝑍)) < ((𝐹‘𝑏) + (𝐶 / (𝑛 · 𝑍)))))
40	36, 37	oveq12d 6031	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑛 · 𝑍) → ((𝐹‘𝑎) + (𝐶 / 𝑎)) = ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))
41	40	breq2d 4098	. . . . . . . . . . . 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 2744	. . . . . . . . . 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 5635	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
46	45	oveq1d 6028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) + (𝐶 / 𝑛)) = ((𝐹‘𝑏) + (𝐶 / 𝑛)))
47	46	breq2d 4098	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (𝐶 / 𝑛)) ↔ (𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛))))
48	45	breq1d 4096	. . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 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 2536	. . . . . . . . . . . . 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 5635	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → (ℤ≥‘𝑛) = (ℤ≥‘𝑎))
53		fveq2 5635	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → (𝐹‘𝑛) = (𝐹‘𝑎))
54		oveq2 6021	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑎 → (𝐶 / 𝑛) = (𝐶 / 𝑎))
55	54	oveq2d 6029	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑏) + (𝐶 / 𝑛)) = ((𝐹‘𝑏) + (𝐶 / 𝑎)))
56	53, 55	breq12d 4099	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) < ((𝐹‘𝑏) + (𝐶 / 𝑛)) ↔ (𝐹‘𝑎) < ((𝐹‘𝑏) + (𝐶 / 𝑎))))
57	53, 54	oveq12d 6031	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) + (𝐶 / 𝑛)) = ((𝐹‘𝑎) + (𝐶 / 𝑎)))
58	57	breq2d 4098	. . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . 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 2913	. . . . . . . . 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 9761	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘𝑛) → 𝑛 ≤ 𝑘)
67	66	adantl 277	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑛 ≤ 𝑘)
68	1	nnred 9149	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑛 ∈ ℝ)
69	15	nnred 9149	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℝ)
70	5	nnrpd 9922	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑍 ∈ ℝ+)
71	68, 69, 70	lemul1d 9968	. . . . . . . . . . 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 9594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝑛 · 𝑍) ∈ ℤ)
74	16	nnzd 9594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝑘 · 𝑍) ∈ ℤ)
75		eluz 9762	. . . . . . . . . . 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 2913	. . . . . . . 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 6028	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) = ((𝐹‘(𝑘 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))
80	79	breq2d 4098	. . . . . . . . 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 4096	. . . . . . . . 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 4096	. . . . . . . 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 6028	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍))) = ((𝐹‘(𝑛 · 𝑍)) + (𝐶 / (𝑛 · 𝑍))))
86	85	breq2d 4098	. . . . . . . 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 9149	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑍 ∈ ℝ)
91	1	nnrpd 9922	. . . . . . . 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 9980	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐶 · 𝑛) < (𝑍 · 𝑛))
95	6	nncnd 9150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝑛 · 𝑍) ∈ ℂ)
96	95	mulid2d 8191	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1 · (𝑛 · 𝑍)) = (𝑛 · 𝑍))
97	96	breq2d 4098	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐶 · 𝑛) < (1 · (𝑛 · 𝑍)) ↔ (𝐶 · 𝑛) < (𝑛 · 𝑍)))
98		1red 8187	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 1 ∈ ℝ)
99	6	nnrpd 9922	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝑛 · 𝑍) ∈ ℝ+)
100	25, 91, 98, 99	lt2mul2divd 9993	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐶 · 𝑛) < (1 · (𝑛 · 𝑍)) ↔ (𝐶 / (𝑛 · 𝑍)) < (1 / 𝑛)))
101	1	nncnd 9150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑛 ∈ ℂ)
102	5	nncnd 9150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑍 ∈ ℂ)
103	101, 102	mulcomd 8194	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝑛 · 𝑍) = (𝑍 · 𝑛))
104	103	breq2d 4098	. . . . . . . 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 8595	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐺‘𝑘) + (𝐶 / (𝑛 · 𝑍))) < ((𝐺‘𝑘) + (1 / 𝑛)))
108	13, 27, 29, 89, 107	lttrd 8298	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)))
109	13, 26	readdcld 8202	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍))) ∈ ℝ)
110	13, 28	readdcld 8202	. . . . 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 8595	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((𝐺‘𝑛) + (𝐶 / (𝑛 · 𝑍))) < ((𝐺‘𝑛) + (1 / 𝑛)))
113	22, 109, 110, 111, 112	lttrd 8298	. . . 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 2603	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛))))
116	115	ralrimiva 2603	1 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((𝐺‘𝑛) < ((𝐺‘𝑘) + (1 / 𝑛)) ∧ (𝐺‘𝑘) < ((𝐺‘𝑛) + (1 / 𝑛))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508   class class class wbr 4086   ↦ cmpt 4148  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013  ℝcr 8024  1c1 8026   + caddc 8028   · cmul 8030   < clt 8207   ≤ cle 8208   / cdiv 8845  ℕcn 9136  ℤcz 9472  ℤ_≥cuz 9748  ℝ⁺crp 9881

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-n0 9396  df-z 9473  df-uz 9749  df-rp 9882

This theorem is referenced by:  cvg1nlemres  11539