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Theorem iscau4 25259

Description: Express the property "𝐹 is a Cauchy sequence of metric 𝐷 " using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)

**Hypotheses**
Ref	Expression
iscau3.2	⊢ 𝑍 = (ℤ_≥‘𝑀)
iscau3.3	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
iscau3.4	⊢ (𝜑 → 𝑀 ∈ ℤ)
iscau4.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
iscau4.6	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = 𝐵)

**Assertion**
Ref	Expression
iscau4	⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))

Distinct variable groups: 𝑗,𝑘,𝑥,𝐷 𝑗,𝐹,𝑘,𝑥 𝜑,𝑗,𝑘,𝑥 𝑗,𝑋,𝑘,𝑥 𝑗,𝑀 𝑗,𝑍,𝑘,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑗,𝑘) 𝐵(𝑥,𝑗,𝑘) 𝑀(𝑥,𝑘)

Proof of Theorem iscau4 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscau3.2	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		iscau3.3	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
3		iscau3.4	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	1, 2, 3	iscau3 25258	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))
5		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
6	5, 1	eleqtrdi 2847	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
7		eluzelz 12792	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
8		uzid 12797	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
9	6, 7, 8	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
10		fveq2 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑗))
11		fveq2 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
12	11	oveq1d 7376	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚)))
13	12	breq1d 5096	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
14	10, 13	raleqbidv 3312	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
15	14	rspcv 3561	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
16	9, 15	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
17	16	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
18		fveq2 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
19	18	oveq2d 7377	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) = ((𝐹‘𝑗)𝐷(𝐹‘𝑘)))
20	19	breq1d 5096	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥))
21	20	cbvralvw 3216	. . . . . . . . . . . 12 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥)
22		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝐹‘𝑘) ∈ 𝑋)
23	22	ralimi 3075	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ 𝑋)
24	11	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑗) ∈ 𝑋))
25	24	rspcv 3561	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ 𝑋 → (𝐹‘𝑗) ∈ 𝑋))
26	9, 23, 25	syl2im 40	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝐹‘𝑗) ∈ 𝑋))
27	26	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑗) ∈ 𝑋)
28		r19.26 3098	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥) ↔ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥))
29	2	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
30		simplr 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑗) ∈ 𝑋)
31		simprr 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑘) ∈ 𝑋)
32		xmetsym 24325	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑗) ∈ 𝑋 ∧ (𝐹‘𝑘) ∈ 𝑋) → ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗)))
33	29, 30, 31, 32	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗)))
34	33	breq1d 5096	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
35	34	biimpd 229	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥 → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
36	35	expimpd 453	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) → (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
37	36	ralimdv 3152	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
38	28, 37	biimtrrid 243	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) → ((∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
39	38	expd 415	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
40	39	impancom 451	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → ((𝐹‘𝑗) ∈ 𝑋 → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
41	27, 40	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
42	21, 41	biimtrid 242	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
43	17, 42	syld 47	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
44	43	imdistanda 571	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
45		r19.26 3098	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
46		r19.26 3098	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
47	44, 45, 46	3imtr4g 296	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
48		df-3an 1089	. . . . . . . . 9 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
49	48	ralbii 3084	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
50		df-3an 1089	. . . . . . . . 9 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
51	50	ralbii 3084	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
52	47, 49, 51	3imtr4g 296	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
53	52	reximdva 3151	. . . . . 6 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
54	53	ralimdv 3152	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
55	54	anim2d 613	. . . 4 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)) → (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
56	4, 55	sylbid 240	. . 3 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) → (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
57		uzssz 12803	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
58	1, 57	eqsstri 3969	. . . . . . . 8 ⊢ 𝑍 ⊆ ℤ
59		ssrexv 3992	. . . . . . . 8 ⊢ (𝑍 ⊆ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
60	58, 59	ax-mp 5	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
61	60	ralimi 3075	. . . . . 6 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
62	61	anim2i 618	. . . . 5 ⊢ ((𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) → (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))
63		iscau2 25257	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
64	62, 63	imbitrrid 246	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) → 𝐹 ∈ (Cau‘𝐷)))
65	2, 64	syl 17	. . 3 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) → 𝐹 ∈ (Cau‘𝐷)))
66	56, 65	impbid 212	. 2 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))))
67		simpl 482	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑗 ∈ 𝑍)
68	1	uztrn2 12801	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
69	67, 68	jca 511	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍))
70		iscau4.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
71	70	adantrl 717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (𝐹‘𝑘) = 𝐴)
72	71	eleq1d 2822	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → ((𝐹‘𝑘) ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
73		iscau4.6	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = 𝐵)
74	73	adantrr 718	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (𝐹‘𝑗) = 𝐵)
75	71, 74	oveq12d 7379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) = (𝐴𝐷𝐵))
76	75	breq1d 5096	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥 ↔ (𝐴𝐷𝐵) < 𝑥))
77	72, 76	3anbi23d 1442	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
78	69, 77	sylan2 594	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
79	78	anassrs 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ (𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
80	79	ralbidva 3159	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
81	80	rexbidva 3160	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
82	81	ralbidv 3161	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))
83	82	anbi2d 631	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))
84	66, 83	bitrd 279	1 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑_pm ℂ) ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ⊆ wss 3890 class class class wbr 5086 dom cdm 5625 ‘cfv 6493 (class class class)co 7361 ↑_pm cpm 8768 ℂcc 11030 < clt 11173 ℤcz 12518 ℤ_≥cuz 12782 ℝ⁺crp 12936 ∞Metcxmet 21332 Cauccau 25233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-z 12519 df-uz 12783 df-rp 12937 df-xneg 13057 df-xadd 13058 df-psmet 21339 df-xmet 21340 df-bl 21342 df-cau 25236

This theorem is referenced by: iscauf 25260 cmetcaulem 25268 caures 38098 caushft 38099

W3C validator