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Theorem iscau4 24523

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 24522	. . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
6	5, 1	eleqtrdi 2847	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
7		eluzelz 12671	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
8		uzid 12676	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
9	6, 7, 8	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
10		fveq2 6811	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑗))
11		fveq2 6811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
12	11	oveq1d 7331	. . . . . . . . . . . . . . . 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 3315	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
15	14	rspcv 3565	. . . . . . . . . . . . 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 481	. . . . . . . . . . 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 6811	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
19	18	oveq2d 7332	. . . . . . . . . . . . . 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 3221	. . . . . . . . . . . 12 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥)
22		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝐹‘𝑘) ∈ 𝑋)
23	22	ralimi 3082	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ 𝑋)
24	11	eleq1d 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑗) ∈ 𝑋))
25	24	rspcv 3565	. . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑗) ∈ 𝑋)
28		r19.26 3110	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → 𝐷 ∈ (∞Met‘𝑋))
30		simplr 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑗) ∈ 𝑋)
31		simprr 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑘) ∈ 𝑋)
32		xmetsym 23580	. . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . 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 452	. . . . . . . . . . . . 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 241	. . . . . . . . . . 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 572	. . . . . . . . 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 3110	. . . . . . . . 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 3110	. . . . . . . . 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 295	. . . . . . . 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 1088	. . . . . . . . 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 3092	. . . . . . . 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 1088	. . . . . . . . 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 3092	. . . . . . . 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 295	. . . . . . 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 3161	. . . . . 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 3162	. . . . 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 612	. . . 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 239	. . 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 12682	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
58	1, 57	eqsstri 3964	. . . . . . . 8 ⊢ 𝑍 ⊆ ℤ
59		ssrexv 3997	. . . . . . . 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 3082	. . . . . 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 617	. . . . 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 24521	. . . . 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 245	. . . 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 211	. 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 483	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑗 ∈ 𝑍)
68	1	uztrn2 12680	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
69	67, 68	jca 512	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍))
70		iscau4.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)
71	70	adantrl 713	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (𝐹‘𝑘) = 𝐴)
72	71	eleq1d 2821	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → ((𝐹‘𝑘) ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
73		iscau4.6	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = 𝐵)
74	73	adantrr 714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (𝐹‘𝑗) = 𝐵)
75	71, 74	oveq12d 7334	. . . . . . . . . 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 1438	. . . . . . . 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 593	. . . . . . 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 468	. . . . . 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 3168	. . . . 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 3169	. . . 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 3170	. . 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 629	. 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 278	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3061 ∃wrex 3070 ⊆ wss 3896 class class class wbr 5086 dom cdm 5607 ‘cfv 6465 (class class class)co 7316 ↑_pm cpm 8665 ℂcc 10948 < clt 11088 ℤcz 12398 ℤ_≥cuz 12661 ℝ⁺crp 12809 ∞Metcxmet 20662 Cauccau 24497

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4850 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5170 df-id 5506 df-po 5520 df-so 5521 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-1st 7877 df-2nd 7878 df-er 8547 df-map 8666 df-pm 8667 df-en 8783 df-dom 8784 df-sdom 8785 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-div 11712 df-2 12115 df-z 12399 df-uz 12662 df-rp 12810 df-xneg 12927 df-xadd 12928 df-psmet 20669 df-xmet 20670 df-bl 20672 df-cau 24500

This theorem is referenced by: iscauf 24524 cmetcaulem 24532 caures 35995 caushft 35996

W3C validator