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Theorem iscau4 25179

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 25178	. . . 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 2838	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
7		eluzelz 12803	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
8		uzid 12808	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
9	6, 7, 8	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
10		fveq2 6858	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑗))
11		fveq2 6858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
12	11	oveq1d 7402	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚)))
13	12	breq1d 5117	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
14	10, 13	raleqbidv 3319	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥))
15	14	rspcv 3584	. . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
19	18	oveq2d 7403	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) = ((𝐹‘𝑗)𝐷(𝐹‘𝑘)))
20	19	breq1d 5117	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < 𝑥 ↔ ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) < 𝑥))
21	20	cbvralvw 3215	. . . . . . . . . . . 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 3066	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) ∈ 𝑋)
24	11	eleq1d 2813	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑗) ∈ 𝑋))
25	24	rspcv 3584	. . . . . . . . . . . . . . 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 3091	. . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑗) ∈ 𝑋)
31		simprr 772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹‘𝑗) ∈ 𝑋) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋)) → (𝐹‘𝑘) ∈ 𝑋)
32		xmetsym 24235	. . . . . . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . . . 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 3091	. . . . . . . . 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 3091	. . . . . . . . 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 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 3075	. . . . . . . 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 3075	. . . . . . . 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 3146	. . . . . 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 3147	. . . . 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 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 12814	. . . . . . . . 9 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
58	1, 57	eqsstri 3993	. . . . . . . 8 ⊢ 𝑍 ⊆ ℤ
59		ssrexv 4016	. . . . . . . 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 3066	. . . . . 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 25177	. . . . 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 12812	. . . . . . . . 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 716	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → (𝐹‘𝑘) = 𝐴)
72	71	eleq1d 2813	. . . . . . . . 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 7405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ 𝑍)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) = (𝐴𝐷𝐵))
76	75	breq1d 5117	. . . . . . . . 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 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 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 3154	. . . . 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 3155	. . . 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 3156	. . 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 630	. 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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ⊆ wss 3914 class class class wbr 5107 dom cdm 5638 ‘cfv 6511 (class class class)co 7387 ↑_pm cpm 8800 ℂcc 11066 < clt 11208 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 ∞Metcxmet 21249 Cauccau 25153

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-z 12530 df-uz 12794 df-rp 12952 df-xneg 13072 df-xadd 13073 df-psmet 21256 df-xmet 21257 df-bl 21259 df-cau 25156

This theorem is referenced by: iscauf 25180 cmetcaulem 25188 caures 37754 caushft 37755

W3C validator