Theorem metss 24521

Description: Two ways of saying that metric 𝐷 generates a finer topology than metric 𝐶. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)

Hypotheses

Ref	Expression
metequiv.3	⊢ 𝐽 = (MetOpen‘𝐶)
metequiv.4	⊢ 𝐾 = (MetOpen‘𝐷)

Assertion

Ref	Expression
metss	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))

Distinct variable groups:   𝑠,𝑟,𝑥,𝐶   𝐽,𝑟,𝑠,𝑥   𝐾,𝑟,𝑠,𝑥   𝐷,𝑟,𝑠,𝑥   𝑋,𝑟,𝑠,𝑥

Proof of Theorem metss

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metequiv.3	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐶)
2	1	mopnval 24448	. . . 4 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐶)))
3	2	adantr 480	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝐽 = (topGen‘ran (ball‘𝐶)))
4		metequiv.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
5	4	mopnval 24448	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
6	5	adantl 481	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝐾 = (topGen‘ran (ball‘𝐷)))
7	3, 6	sseq12d 4017	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ (topGen‘ran (ball‘𝐶)) ⊆ (topGen‘ran (ball‘𝐷))))
8		blbas 24440	. . 3 ⊢ (𝐶 ∈ (∞Met‘𝑋) → ran (ball‘𝐶) ∈ TopBases)
9		unirnbl 24430	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐶) = 𝑋)
10	9	adantr 480	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐶) = 𝑋)
11		unirnbl 24430	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
12	11	adantl 481	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐷) = 𝑋)
13	10, 12	eqtr4d 2780	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐶) = ∪ ran (ball‘𝐷))
14		tgss2 22994	. . 3 ⊢ ((ran (ball‘𝐶) ∈ TopBases ∧ ∪ ran (ball‘𝐶) = ∪ ran (ball‘𝐷)) → ((topGen‘ran (ball‘𝐶)) ⊆ (topGen‘ran (ball‘𝐷)) ↔ ∀𝑥 ∈ ∪ ran (ball‘𝐶)∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
15	8, 13, 14	syl2an2r 685	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ((topGen‘ran (ball‘𝐶)) ⊆ (topGen‘ran (ball‘𝐷)) ↔ ∀𝑥 ∈ ∪ ran (ball‘𝐶)∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
16	10	raleqdv 3326	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥 ∈ ∪ ran (ball‘𝐶)∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))))
17		blssex 24437	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
18	17	adantll 714	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
19	18	imbi2d 340	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ (𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)))
20	19	ralbidv 3178	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)))
21		rpxr 13044	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
22		blelrn 24427	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
23	21, 22	syl3an3 1166	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
24		blcntr 24423	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)𝑟))
25		eleq2 2830	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑥(ball‘𝐶)𝑟)))
26		sseq2 4010	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → ((𝑥(ball‘𝐷)𝑠) ⊆ 𝑦 ↔ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
27	26	rexbidv 3179	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → (∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦 ↔ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
28	25, 27	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → ((𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦) ↔ (𝑥 ∈ (𝑥(ball‘𝐶)𝑟) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟))))
29	28	rspcv 3618	. . . . . . . . . 10 ⊢ ((𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶) → (∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦) → (𝑥 ∈ (𝑥(ball‘𝐶)𝑟) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟))))
30	29	com23 86	. . . . . . . . 9 ⊢ ((𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶) → (𝑥 ∈ (𝑥(ball‘𝐶)𝑟) → (∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟))))
31	23, 24, 30	sylc 65	. . . . . . . 8 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
32	31	ad4ant134 1175	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
33	32	ralrimdva 3154	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦) → ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
34		blss 24435	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
35	34	3expb 1121	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
36	35	ad4ant14 752	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
37		r19.29 3114	. . . . . . . . . . . 12 ⊢ ((∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦) → ∃𝑟 ∈ ℝ+ (∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦))
38		sstr 3992	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦) → (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)
39	38	expcom 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑥(ball‘𝐶)𝑟) ⊆ 𝑦 → ((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
40	39	reximdv 3170	. . . . . . . . . . . . . 14 ⊢ ((𝑥(ball‘𝐶)𝑟) ⊆ 𝑦 → (∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
41	40	impcom 407	. . . . . . . . . . . . 13 ⊢ ((∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)
42	41	rexlimivw 3151	. . . . . . . . . . . 12 ⊢ (∃𝑟 ∈ ℝ+ (∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)
43	37, 42	syl 17	. . . . . . . . . . 11 ⊢ ((∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)
44	43	ex 412	. . . . . . . . . 10 ⊢ (∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → (∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
45	36, 44	syl5com 31	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦)) → (∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
46	45	expr 456	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ ran (ball‘𝐶)) → (𝑥 ∈ 𝑦 → (∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)))
47	46	com23 86	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ ran (ball‘𝐶)) → (∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → (𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)))
48	47	ralrimdva 3154	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → ∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)))
49	33, 48	impbid 212	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦) ↔ ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
50	20, 49	bitrd 279	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
51	50	ralbidva 3176	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
52	16, 51	bitrd 279	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥 ∈ ∪ ran (ball‘𝐶)∀𝑦 ∈ ran (ball‘𝐶)(𝑥 ∈ 𝑦 → ∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
53	7, 15, 52	3bitrd 305	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∪ cuni 4907  ran crn 5686  ‘cfv 6561  (class class class)co 7431  ℝ^*cxr 11294  ℝ⁺crp 13034  topGenctg 17482  ∞Metcxmet 21349  ballcbl 21351  MetOpencmopn 21354  TopBasesctb 22952

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-bl 21359  df-mopn 21360  df-bases 22953

This theorem is referenced by:  metequiv  24522  metss2  24525