Theorem metss 24740

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 24670	. . . 4 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐶)))
3	2	adantr 486	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝐽 = (topGen‘ran (ball‘𝐶)))
4		metequiv.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
5	4	mopnval 24670	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
6	5	adantl 487	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝐾 = (topGen‘ran (ball‘𝐷)))
7	3, 6	sseq12d 3967	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ (topGen‘ran (ball‘𝐶)) ⊆ (topGen‘ran (ball‘𝐷))))
8		blbas 24662	. . 3 ⊢ (𝐶 ∈ (∞Met‘𝑋) → ran (ball‘𝐶) ∈ TopBases)
9		unirnbl 24652	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐶) = 𝑋)
10	9	adantr 486	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐶) = 𝑋)
11		unirnbl 24652	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
12	11	adantl 487	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐷) = 𝑋)
13	10, 12	eqtr4d 2800	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐶) = ∪ ran (ball‘𝐷))
14		tgss2 23218	. . 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 698	. 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 3321	. . 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 24659	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
18	17	adantll 727	. . . . . . 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 343	. . . . . 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 3187	. . . . 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 13056	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
22		blelrn 24649	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
23	21, 22	syl3an3 1183	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
24		blcntr 24645	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)𝑟))
25		eleq2 2851	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑥(ball‘𝐶)𝑟)))
26		sseq2 3960	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → ((𝑥(ball‘𝐷)𝑠) ⊆ 𝑦 ↔ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
27	26	rexbidv 3188	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 3575	. . . . . . . . . 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 87	. . . . . . . . 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 66	. . . . . . . 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 1193	. . . . . . 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 3164	. . . . . 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 24657	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
35	34	3expb 1138	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
36	35	ad4ant14 765	. . . . . . . . . 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 3127	. . . . . . . . . . . 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 3942	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦) → (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)
39	38	expcom 419	. . . . . . . . . . . . . . 15 ⊢ ((𝑥(ball‘𝐶)𝑟) ⊆ 𝑦 → ((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
40	39	reximdv 3179	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 3161	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 418	. . . . . . . . . 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 32	. . . . . . . . 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 462	. . . . . . . 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 87	. . . . . . 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 3164	. . . . . 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 215	. . . . 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 282	. . . 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 3185	. . 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 282	. 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 308	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902  ∪ cuni 4870  ran crn 5660  ‘cfv 6537  (class class class)co 7417  ℝ^*cxr 11270  ℝ⁺crp 13046  topGenctg 17528  ∞Metcxmet 21576  ballcbl 21578  MetOpencmopn 21581  TopBasesctb 23176

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-sup 9416  df-inf 9417  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-n0 12533  df-z 12620  df-uz 12892  df-q 13002  df-rp 13047  df-xneg 13167  df-xadd 13168  df-xmul 13169  df-topgen 17534  df-psmet 21583  df-xmet 21584  df-bl 21586  df-mopn 21587  df-bases 23177

This theorem is used by:  metequiv  24741  metss2  24744