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Theorem iscau4 24130

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 24129	. . . 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 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
6	5, 1	eleqtrdi 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
7		eluzelz 12413	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
8		uzid 12418	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
9	6, 7, 8	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
10		fveq2 6695	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑗))
11		fveq2 6695	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
12	11	oveq1d 7206	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚)))
13	12	breq1d 5049	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
14	10, 13	raleqbidv 3303	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
15	14	rspcv 3522	. . . . . . . . . . . . 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 484	. . . . . . . . . . 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 6695	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
19	18	oveq2d 7207	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) = ((𝐹‘𝑗)𝐷(𝐹‘𝑘)))
20	19	breq1d 5049	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥))
21	20	cbvralvw 3348	. . . . . . . . . . . 12 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥)
22		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝐹‘𝑘) ∈ 𝑋)
23	22	ralimi 3073	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ 𝑋)
24	11	eleq1d 2815	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑗) ∈ 𝑋))
25	24	rspcv 3522	. . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑗) ∈ 𝑋)
28		r19.26 3082	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 23199	. . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . 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 5049	. . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . . 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 3091	. . . . . . . . . . . . . . . 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	syl5bir 246	. . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . 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 455	. . . . . . . . . . . . 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	syl5bi 245	. . . . . . . . . . 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 575	. . . . . . . . 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 3082	. . . . . . . . 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 3082	. . . . . . . . 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 299	. . . . . . . 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 1091	. . . . . . . . 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 3078	. . . . . . . 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 1091	. . . . . . . . 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 3078	. . . . . . . 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 299	. . . . . . 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 3183	. . . . . 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 3091	. . . . 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 615	. . . 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 243	. . 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 12424	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
58	1, 57	eqsstri 3921	. . . . . . . 8 ⊢ 𝑍 ⊆ ℤ
59		ssrexv 3954	. . . . . . . 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 3073	. . . . . 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 620	. . . . 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 24128	. . . . 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	syl5ibr 249	. . . 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 215	. 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 486	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑗 ∈ 𝑍)
68	1	uztrn2 12422	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
69	67, 68	jca 515	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍))
70		iscau4.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
71	70	adantrl 716	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (𝐹‘𝑘) = 𝐴)
72	71	eleq1d 2815	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → ((𝐹‘𝑘) ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
73		iscau4.6	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = 𝐵)
74	73	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (𝐹‘𝑗) = 𝐵)
75	71, 74	oveq12d 7209	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) = (𝐴𝐷𝐵))
76	75	breq1d 5049	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥 ↔ (𝐴𝐷𝐵) < 𝑥))
77	72, 76	3anbi23d 1441	. . . . . . . 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 596	. . . . . . 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 471	. . . . . 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 3107	. . . . 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 3205	. . . 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 3108	. . 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 632	. 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 282	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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ⊆ wss 3853 class class class wbr 5039 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 ↑_pm cpm 8487 ℂcc 10692 < clt 10832 ℤcz 12141 ℤ_≥cuz 12403 ℝ⁺crp 12551 ∞Metcxmet 20302 Cauccau 24104

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-2 11858 df-z 12142 df-uz 12404 df-rp 12552 df-xneg 12669 df-xadd 12670 df-psmet 20309 df-xmet 20310 df-bl 20312 df-cau 24107

This theorem is referenced by: iscauf 24131 cmetcaulem 24139 caures 35604 caushft 35605

W3C validator