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Theorem iscau3 25507

Description: Express the Cauchy sequence property in the more conventional three-quantifier form. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)

**Hypotheses**
Ref	Expression
iscau3.2	⊢ 𝑍 = (ℤ_≥‘𝑀)
iscau3.3	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
iscau3.4	⊢ (𝜑 → 𝑀 ∈ ℤ)

**Assertion**
Ref	Expression
iscau3	⊢ (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))))

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

**Proof of Theorem iscau3**
Step	Hyp	Ref	Expression
1		iscau3.3	. . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
2		iscau2 25506	. . 3 ⊢ (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))))
3	1, 2	syl 18	. 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))))
4	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑_pm ℂ)) → 𝐷 ∈ (∞Met‘𝑋))
5		ssid 3953	. . . . . . 7 ⊢ ℤ ⊆ ℤ
6		simpr 490	. . . . . . 7 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝐹‘𝑘) ∈ 𝑋)
7		eleq1 2848	. . . . . . 7 ⊢ ((𝐹‘𝑘) = (𝐹‘𝑗) → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑗) ∈ 𝑋))
8		eleq1 2848	. . . . . . 7 ⊢ ((𝐹‘𝑘) = (𝐹‘𝑚) → ((𝐹‘𝑘) ∈ 𝑋 ↔ (𝐹‘𝑚) ∈ 𝑋))
9		xmetsym 24574	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑗) ∈ 𝑋 ∧ (𝐹‘𝑘) ∈ 𝑋) → ((𝐹‘𝑗)𝐷(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗)))
10	9	fveq2d 6883	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑗) ∈ 𝑋 ∧ (𝐹‘𝑘) ∈ 𝑋) → ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑘))) = ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))))
11		xmetsym 24574	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑚) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋) → ((𝐹‘𝑚)𝐷(𝐹‘𝑗)) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚)))
12	11	fveq2d 6883	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑚) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋) → ( I ‘((𝐹‘𝑚)𝐷(𝐹‘𝑗))) = ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))))
13		simp1 1154	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → 𝐷 ∈ (∞Met‘𝑋))
14		simp2l 1218	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝐹‘𝑘) ∈ 𝑋)
15		simp3l 1220	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝐹‘𝑗) ∈ 𝑋)
16		xmetcl 24558	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ^*)
17	13, 14, 15, 16	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ^*)
18		simp2r 1219	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝐹‘𝑚) ∈ 𝑋)
19		xmetcl 24558	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑗) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) → ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ ℝ^*)
20	13, 15, 18, 19	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ ℝ^*)
21		simp3r 1221	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → 𝑥 ∈ ℝ)
22	21	rehalfcld 12516	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝑥 / 2) ∈ ℝ)
23	22	rexrd 11284	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (𝑥 / 2) ∈ ℝ^*)
24		xlt2add 13313	. . . . . . . . . 10 ⊢ (((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ^* ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ ℝ^) ∧ ((𝑥 / 2) ∈ ℝ^ ∧ (𝑥 / 2) ∈ ℝ^*)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < ((𝑥 / 2) +_𝑒 (𝑥 / 2))))
25	17, 20, 23, 23, 24	syl22anc 852	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < ((𝑥 / 2) +_𝑒 (𝑥 / 2))))
26	22, 22	rexaddd 13287	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝑥 / 2) +_𝑒 (𝑥 / 2)) = ((𝑥 / 2) + (𝑥 / 2)))
27	21	recnd 11262	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → 𝑥 ∈ ℂ)
28	27	2halvesd 12515	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝑥 / 2) + (𝑥 / 2)) = 𝑥)
29	26, 28	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝑥 / 2) +_𝑒 (𝑥 / 2)) = 𝑥)
30	29	breq2d 5115	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < ((𝑥 / 2) +_𝑒 (𝑥 / 2)) ↔ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < 𝑥))
31		xmettri 24578	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))))
32	13, 14, 18, 15, 31	syl13anc 1399	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))))
33		xmetcl 24558	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ ℝ^*)
34	13, 14, 18, 33	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ ℝ^*)
35	17, 20	xaddcld 13354	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) ∈ ℝ^*)
36	21	rexrd 11284	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → 𝑥 ∈ ℝ^*)
37		xrlelttr 13208	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ ℝ^* ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < 𝑥) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
38	34, 35, 36, 37	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ≤ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) ∧ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < 𝑥) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
39	32, 38	mpand 708	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < 𝑥 → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
40	30, 39	sylbid 243	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) +_𝑒 ((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < ((𝑥 / 2) +_𝑒 (𝑥 / 2)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
41	25, 40	syld 48	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
42		ovex 7447	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ V
43		fvi 6955	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ V → ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗)))
44	42, 43	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗))
45	44	breq1i 5110	. . . . . . . . 9 ⊢ (( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < (𝑥 / 2) ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2))
46		ovex 7447	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ V
47		fvi 6955	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑗)𝐷(𝐹‘𝑚)) ∈ V → ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚)))
48	46, 47	ax-mp 5	. . . . . . . . . 10 ⊢ ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) = ((𝐹‘𝑗)𝐷(𝐹‘𝑚))
49	48	breq1i 5110	. . . . . . . . 9 ⊢ (( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < (𝑥 / 2) ↔ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2))
50	45, 49	anbi12i 640	. . . . . . . 8 ⊢ ((( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < (𝑥 / 2) ∧ ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < (𝑥 / 2)) ↔ (((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < (𝑥 / 2) ∧ ((𝐹‘𝑗)𝐷(𝐹‘𝑚)) < (𝑥 / 2)))
51		ovex 7447	. . . . . . . . . 10 ⊢ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ V
52		fvi 6955	. . . . . . . . . 10 ⊢ (((𝐹‘𝑘)𝐷(𝐹‘𝑚)) ∈ V → ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) = ((𝐹‘𝑘)𝐷(𝐹‘𝑚)))
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) = ((𝐹‘𝑘)𝐷(𝐹‘𝑚))
54	53	breq1i 5110	. . . . . . . 8 ⊢ (( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)
55	41, 50, 54	3imtr4g 299	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑚) ∈ 𝑋) ∧ ((𝐹‘𝑗) ∈ 𝑋 ∧ 𝑥 ∈ ℝ)) → ((( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < (𝑥 / 2) ∧ ( I ‘((𝐹‘𝑗)𝐷(𝐹‘𝑚))) < (𝑥 / 2)) → ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥))
56	5, 6, 7, 8, 10, 12, 55	cau3lem 15443	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥)))
57	4, 56	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑_pm ℂ)) → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥)))
58	44	breq1i 5110	. . . . . . . . . 10 ⊢ (( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥 ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)
59	58	anbi2i 635	. . . . . . . . 9 ⊢ (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
60		df-3an 1105	. . . . . . . . 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))
61	59, 60	bitr4i 281	. . . . . . . 8 ⊢ (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
62	61	ralbii 3108	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
63	62	rexbii 3109	. . . . . 6 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑗))) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥))
64	63	ralbii 3108	. . . . 5 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ( I ‘((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))
65	54	ralbii 3108	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)
66	65	anbi2i 635	. . . . . . . . 9 ⊢ (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
67		df-3an 1105	. . . . . . . . 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))
68	66, 67	bitr4i 281	. . . . . . . 8 ⊢ (((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
69	68	ralbii 3108	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
70	69	rexbii 3109	. . . . . 6 ⊢ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
71	70	ralbii 3108	. . . . 5 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)( I ‘((𝐹‘𝑘)𝐷(𝐹‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))
72	57, 64, 71	3bitr3g 316	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑_pm ℂ)) → (∀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 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
73		iscau3.4	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
74	73	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑_pm ℂ)) → 𝑀 ∈ ℤ)
75		iscau3.2	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
76	75	rexuz3 15437	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (∃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 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
77	74, 76	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝑋 ↑_pm ℂ)) → (∃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 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
78	77	ralbidv 3185	. . . 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) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
79	72, 78	bitr4d 285	. . 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) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))
80	79	pm5.32da 590	. 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 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥))))
81	3, 80	bitrd 282	1 ⊢ (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))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 Vcvv 3450 class class class wbr 5103 I cid 5549 dom cdm 5655 ‘cfv 6533 (class class class)co 7414 ↑_pm cpm 8828 ℂcc 11123 ℝcr 11124 + caddc 11128 ℝ^*cxr 11267 < clt 11268 ≤ cle 11269 / cdiv 11896 2c2 12320 ℤcz 12616 ℤ_≥cuz 12888 ℝ⁺crp 13043 +_𝑒 cxad 13162 ∞Metcxmet 21571 Cauccau 25482

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8697 df-map 8829 df-pm 8830 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-z 12617 df-uz 12889 df-rp 13044 df-xneg 13164 df-xadd 13165 df-psmet 21578 df-xmet 21579 df-bl 21581 df-cau 25485

This theorem is used by: iscau4 25508 caucfil 25512 cmetcaulem 25517 heibor1lem 38560

W3C validator