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Theorem iscfil3 25194

Description: A filter is Cauchy iff it contains a ball of any chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)

**Assertion**
Ref	Expression
iscfil3	⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)))

Distinct variable groups: 𝑥,𝑟,𝐹 𝑋,𝑟,𝑥 𝐷,𝑟,𝑥

Proof of Theorem iscfil3 Dummy variables 𝑢 𝑠 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfilfil 25188	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))
2		cfil3i 25190	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
3	2	3expa 1116	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
4	3	ralrimiva 3142	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
5	1, 4	jca 511	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹))
6		simprl 770	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
7		rphalfcl 13027	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ⁺ → (𝑠 / 2) ∈ ℝ⁺)
8	7	adantl 481	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) → (𝑠 / 2) ∈ ℝ⁺)
9		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑟 = (𝑠 / 2) → (𝑥(ball‘𝐷)𝑟) = (𝑥(ball‘𝐷)(𝑠 / 2)))
10	9	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 / 2) → ((𝑥(ball‘𝐷)𝑟) ∈ 𝐹 ↔ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹))
11	10	rexbidv 3174	. . . . . . . 8 ⊢ (𝑟 = (𝑠 / 2) → (∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹 ↔ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹))
12	11	rspcv 3604	. . . . . . 7 ⊢ ((𝑠 / 2) ∈ ℝ⁺ → (∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹 → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹))
13	8, 12	syl 17	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) → (∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹 → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹))
14		simprr 772	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) → (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)
15		simp-4l 782	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝐷 ∈ (∞Met‘𝑋))
16		simplrl 776	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑥 ∈ 𝑋)
17		simpllr 775	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑠 ∈ ℝ⁺)
18	17	rpred 13042	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑠 ∈ ℝ)
19		simprl 770	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))
20		blhalf 24304	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑠 ∈ ℝ ∧ 𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → (𝑥(ball‘𝐷)(𝑠 / 2)) ⊆ (𝑢(ball‘𝐷)𝑠))
21	15, 16, 18, 19, 20	syl22anc 838	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → (𝑥(ball‘𝐷)(𝑠 / 2)) ⊆ (𝑢(ball‘𝐷)𝑠))
22		simprr 772	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))
23	21, 22	sseldd 3979	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑣 ∈ (𝑢(ball‘𝐷)𝑠))
24	17	rpxrd 13043	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑠 ∈ ℝ^*)
25	17, 7	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → (𝑠 / 2) ∈ ℝ⁺)
26	25	rpxrd 13043	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → (𝑠 / 2) ∈ ℝ^*)
27		blssm 24317	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑠 / 2) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑠 / 2)) ⊆ 𝑋)
28	15, 16, 26, 27	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → (𝑥(ball‘𝐷)(𝑠 / 2)) ⊆ 𝑋)
29	28, 19	sseldd 3979	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑢 ∈ 𝑋)
30	28, 22	sseldd 3979	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → 𝑣 ∈ 𝑋)
31		elbl2 24289	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠 ∈ ℝ^*) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑣 ∈ (𝑢(ball‘𝐷)𝑠) ↔ (𝑢𝐷𝑣) < 𝑠))
32	15, 24, 29, 30, 31	syl22anc 838	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → (𝑣 ∈ (𝑢(ball‘𝐷)𝑠) ↔ (𝑢𝐷𝑣) < 𝑠))
33	23, 32	mpbid 231	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) ∧ (𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)) ∧ 𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2)))) → (𝑢𝐷𝑣) < 𝑠)
34	33	ralrimivva 3196	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) → ∀𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2))∀𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2))(𝑢𝐷𝑣) < 𝑠)
35		raleq 3318	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥(ball‘𝐷)(𝑠 / 2)) → (∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠 ↔ ∀𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2))(𝑢𝐷𝑣) < 𝑠))
36	35	raleqbi1dv 3329	. . . . . . . . 9 ⊢ (𝑦 = (𝑥(ball‘𝐷)(𝑠 / 2)) → (∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠 ↔ ∀𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2))∀𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2))(𝑢𝐷𝑣) < 𝑠))
37	36	rspcev 3608	. . . . . . . 8 ⊢ (((𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹 ∧ ∀𝑢 ∈ (𝑥(ball‘𝐷)(𝑠 / 2))∀𝑣 ∈ (𝑥(ball‘𝐷)(𝑠 / 2))(𝑢𝐷𝑣) < 𝑠) → ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠)
38	14, 34, 37	syl2anc 583	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹)) → ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠)
39	38	rexlimdvaa 3152	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) → (∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)(𝑠 / 2)) ∈ 𝐹 → ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠))
40	13, 39	syld 47	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) → (∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹 → ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠))
41	40	ralrimdva 3150	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹 → ∀𝑠 ∈ ℝ⁺ ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠))
42	41	impr 454	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)) → ∀𝑠 ∈ ℝ⁺ ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠)
43		iscfil2 25187	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ ℝ⁺ ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠)))
44	43	adantr 480	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ ℝ⁺ ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠)))
45	6, 42, 44	mpbir2and 712	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)) → 𝐹 ∈ (CauFil‘𝐷))
46	5, 45	impbida 800	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 ⊆ wss 3945 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ℝcr 11131 ℝ^*cxr 11271 < clt 11272 / cdiv 11895 2c2 12291 ℝ⁺crp 13000 ∞Metcxmet 21257 ballcbl 21259 Filcfil 23742 CauFilccfil 25173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-2 12299 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ico 13356 df-psmet 21264 df-xmet 21265 df-bl 21267 df-fbas 21269 df-fil 23743 df-cfil 25176

This theorem is referenced by: equivcfil 25220 flimcfil 25235

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