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Theorem iscfil3 25219

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 25213	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))
2		cfil3i 25215	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)
3	2	3expa 1115	. . . 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 510	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹))
6		simprl 769	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
7		rphalfcl 13039	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ⁺ → (𝑠 / 2) ∈ ℝ⁺)
8	7	adantl 480	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ ℝ⁺) → (𝑠 / 2) ∈ ℝ⁺)
9		oveq2 7432	. . . . . . . . . 10 ⊢ (𝑟 = (𝑠 / 2) → (𝑥(ball‘𝐷)𝑟) = (𝑥(ball‘𝐷)(𝑠 / 2)))
10	9	eleq1d 2813	. . . . . . . . 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 3605	. . . . . . 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 771	. . . . . . . 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 781	. . . . . . . . . . . 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 775	. . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 13054	. . . . . . . . . . . 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 769	. . . . . . . . . . . 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 24329	. . . . . . . . . . . 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 837	. . . . . . . . . . 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 771	. . . . . . . . . . 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 3981	. . . . . . . . . 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 13055	. . . . . . . . . . 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 13055	. . . . . . . . . . . . 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 24342	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑠 / 2) ∈ ℝ^*) → (𝑥(ball‘𝐷)(𝑠 / 2)) ⊆ 𝑋)
28	15, 16, 26, 27	syl3anc 1368	. . . . . . . . . . . 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 3981	. . . . . . . . . . 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 3981	. . . . . . . . . . 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 24314	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠 ∈ ℝ^*) ∧ (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) → (𝑣 ∈ (𝑢(ball‘𝐷)𝑠) ↔ (𝑢𝐷𝑣) < 𝑠))
32	15, 24, 29, 30, 31	syl22anc 837	. . . . . . . . . 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 3609	. . . . . . . 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 582	. . . . . . 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 453	. . 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 25212	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ ℝ⁺ ∃𝑦 ∈ 𝐹 ∀𝑢 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑢𝐷𝑣) < 𝑠)))
44	43	adantr 479	. . 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 711	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)) → 𝐹 ∈ (CauFil‘𝐷))
46	5, 45	impbida 799	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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3057 ∃wrex 3066 ⊆ wss 3947 class class class wbr 5150 ‘cfv 6551 (class class class)co 7424 ℝcr 11143 ℝ^*cxr 11283 < clt 11284 / cdiv 11907 2c2 12303 ℝ⁺crp 13012 ∞Metcxmet 21269 ballcbl 21271 Filcfil 23767 CauFilccfil 25198

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-po 5592 df-so 5593 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-2 12311 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ico 13368 df-psmet 21276 df-xmet 21277 df-bl 21279 df-fbas 21281 df-fil 23768 df-cfil 25201

This theorem is referenced by: equivcfil 25245 flimcfil 25260

W3C validator