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Theorem caurcvg2 15624

Description: A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.)

**Hypotheses**
Ref	Expression
caucvg.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
caurcvg2.2	⊢ (𝜑 → 𝐹 ∈ 𝑉)
caurcvg2.3	⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))

**Assertion**
Ref	Expression
caurcvg2	⊢ (𝜑 → 𝐹 ∈ dom ⇝ )

Distinct variable groups: 𝑗,𝑘,𝑥,𝐹 𝑗,𝑀,𝑘,𝑥 𝜑,𝑗,𝑘,𝑥 𝑗,𝑍,𝑘,𝑥 Allowed substitution hints: 𝑉(𝑥,𝑗,𝑘)

Proof of Theorem caurcvg2 Dummy variables 𝑖 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1rp 12978	. . . 4 ⊢ 1 ∈ ℝ⁺
2	1	ne0ii 4338	. . 3 ⊢ ℝ⁺ ≠ ∅
3		caurcvg2.3	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
4		r19.2z 4495	. . 3 ⊢ ((ℝ⁺ ≠ ∅ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)) → ∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
5	2, 3, 4	sylancr 588	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))
6		simpl 484	. . . . . 6 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → (𝐹‘𝑘) ∈ ℝ)
7	6	ralimi 3084	. . . . 5 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)
8		eqid 2733	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑗) = (ℤ_≥‘𝑗)
9		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)
10		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
11	10	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ))
12	11	rspccva 3612	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑛) ∈ ℝ)
13	9, 12	sylan 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑛) ∈ ℝ)
14	13	fmpttd 7115	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → (𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛)):(ℤ_≥‘𝑗)⟶ℝ)
15		fveq2 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (ℤ_≥‘𝑗) = (ℤ_≥‘𝑚))
16		fveq2 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚))
17	16	oveq2d 7425	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑚 → ((𝐹‘𝑘) − (𝐹‘𝑗)) = ((𝐹‘𝑘) − (𝐹‘𝑚)))
18	17	fveq2d 6896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑚 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) = (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))))
19	18	breq1d 5159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑚 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥))
20	19	anbi2d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)))
21	15, 20	raleqbidv 3343	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)))
22	21	cbvrexvw 3236	. . . . . . . . . . . . . . 15 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) ↔ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥))
23		fveq2 6892	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → (𝐹‘𝑘) = (𝐹‘𝑖))
24	23	eleq1d 2819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑖) ∈ ℝ))
25	23	fvoveq1d 7431	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) = (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))))
26	25	breq1d 5159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑖 → ((abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥 ↔ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
27	24, 26	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑖 → (((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) ↔ ((𝐹‘𝑖) ∈ ℝ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥)))
28	27	cbvralvw 3235	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) ↔ ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℝ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
29		recn 11200	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑖) ∈ ℝ → (𝐹‘𝑖) ∈ ℂ)
30	29	anim1i 616	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑖) ∈ ℝ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) → ((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
31	30	ralimi 3084	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℝ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) → ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
32	28, 31	sylbi 216	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) → ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
33	32	reximi 3085	. . . . . . . . . . . . . . 15 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
34	22, 33	sylbi 216	. . . . . . . . . . . . . 14 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
35	34	ralimi 3084	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
36	3, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
37	36	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → ∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
38		caucvg.1	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ_≥‘𝑀)
39	38, 8	cau4 15303	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝑍 → (∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ (ℤ_≥‘𝑗)∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥)))
40	39	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → (∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ (ℤ_≥‘𝑗)∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥)))
41	37, 40	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → ∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ (ℤ_≥‘𝑗)∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
42		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) → (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥)
43	8	uztrn2 12841	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → 𝑖 ∈ (ℤ_≥‘𝑗))
44		fveq2 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
45		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛)) = (𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))
46		fvex 6905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹‘𝑖) ∈ V
47	44, 45, 46	fvmpt 6999	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (ℤ_≥‘𝑗) → ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) = (𝐹‘𝑖))
48	43, 47	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) = (𝐹‘𝑖))
49		fveq2 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
50		fvex 6905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹‘𝑚) ∈ V
51	49, 45, 50	fvmpt 6999	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ_≥‘𝑗) → ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚) = (𝐹‘𝑚))
52	51	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚) = (𝐹‘𝑚))
53	48, 52	oveq12d 7427	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → (((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) − ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚)) = ((𝐹‘𝑖) − (𝐹‘𝑚)))
54	53	fveq2d 6896	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → (abs‘(((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) − ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚))) = (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))))
55	54	breq1d 5159	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → ((abs‘(((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) − ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚))) < 𝑥 ↔ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥))
56	42, 55	imbitrrid 245	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑗) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → (((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) → (abs‘(((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) − ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚))) < 𝑥))
57	56	ralimdva 3168	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ_≥‘𝑗) → (∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) → ∀𝑖 ∈ (ℤ_≥‘𝑚)(abs‘(((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) − ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚))) < 𝑥))
58	57	reximia 3082	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ (ℤ_≥‘𝑗)∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) → ∃𝑚 ∈ (ℤ_≥‘𝑗)∀𝑖 ∈ (ℤ_≥‘𝑚)(abs‘(((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) − ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚))) < 𝑥)
59	58	ralimi 3084	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ (ℤ_≥‘𝑗)∀𝑖 ∈ (ℤ_≥‘𝑚)((𝐹‘𝑖) ∈ ℂ ∧ (abs‘((𝐹‘𝑖) − (𝐹‘𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ (ℤ_≥‘𝑗)∀𝑖 ∈ (ℤ_≥‘𝑚)(abs‘(((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) − ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚))) < 𝑥)
60	41, 59	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → ∀𝑥 ∈ ℝ⁺ ∃𝑚 ∈ (ℤ_≥‘𝑗)∀𝑖 ∈ (ℤ_≥‘𝑚)(abs‘(((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑖) − ((𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))‘𝑚))) < 𝑥)
61	8, 14, 60	caurcvg 15623	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → (𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛)) ⇝ (lim sup‘(𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))))
62		eluzelz 12832	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
63	62, 38	eleq2s 2852	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
64	63	ad2antrl 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → 𝑗 ∈ ℤ)
65		caurcvg2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
66	65	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → 𝐹 ∈ 𝑉)
67		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
68	67	cbvmptv 5262	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛)) = (𝑘 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑘))
69	8, 68	climmpt 15515	. . . . . . . . 9 ⊢ ((𝑗 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ (lim sup‘(𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))) ↔ (𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛)) ⇝ (lim sup‘(𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛)))))
70	64, 66, 69	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → (𝐹 ⇝ (lim sup‘(𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))) ↔ (𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛)) ⇝ (lim sup‘(𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛)))))
71	61, 70	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → 𝐹 ⇝ (lim sup‘(𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))))
72		climrel 15436	. . . . . . . 8 ⊢ Rel ⇝
73	72	releldmi 5948	. . . . . . 7 ⊢ (𝐹 ⇝ (lim sup‘(𝑛 ∈ (ℤ_≥‘𝑗) ↦ (𝐹‘𝑛))) → 𝐹 ∈ dom ⇝ )
74	71, 73	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ)) → 𝐹 ∈ dom ⇝ )
75	74	expr 458	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ ℝ → 𝐹 ∈ dom ⇝ ))
76	7, 75	syl5 34	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → 𝐹 ∈ dom ⇝ ))
77	76	rexlimdva 3156	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → 𝐹 ∈ dom ⇝ ))
78	77	rexlimdvw 3161	. 2 ⊢ (𝜑 → (∃𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥) → 𝐹 ∈ dom ⇝ ))
79	5, 78	mpd 15	1 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∅c0 4323 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 1c1 11111 < clt 11248 − cmin 11444 ℤcz 12558 ℤ_≥cuz 12822 ℝ⁺crp 12974 abscabs 15181 lim supclsp 15414 ⇝ cli 15428

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ico 13330 df-fl 13757 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433

This theorem is referenced by: iseralt 15631 cvgcmp 15762

W3C validator