Theorem metss 23645

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 23572	. . . 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 23572	. . . 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 3958	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ (topGen‘ran (ball‘𝐶)) ⊆ (topGen‘ran (ball‘𝐷))))
8		blbas 23564	. . 3 ⊢ (𝐶 ∈ (∞Met‘𝑋) → ran (ball‘𝐶) ∈ TopBases)
9		unirnbl 23554	. . . . 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 23554	. . . . 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 2782	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐶) = ∪ ran (ball‘𝐷))
14		tgss2 22118	. . 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 681	. 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 3346	. . 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 23561	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
18	17	adantll 710	. . . . . . 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 3122	. . . . 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 12721	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
22		blelrn 23551	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
23	21, 22	syl3an3 1163	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
24		blcntr 23547	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)𝑟))
25		eleq2 2828	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑥(ball‘𝐶)𝑟)))
26		sseq2 3951	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → ((𝑥(ball‘𝐷)𝑠) ⊆ 𝑦 ↔ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
27	26	rexbidv 3227	. . . . . . . . . . . 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 3555	. . . . . . . . . 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 1172	. . . . . . 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 3114	. . . . . 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 23559	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
35	34	3expb 1118	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
36	35	ad4ant14 748	. . . . . . . . . 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 3185	. . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . 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 3203	. . . . . . . . . . . . . 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 3212	. . . . . . . . . . . 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 3114	. . . . . 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 211	. . . . 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 278	. . . 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 3121	. . 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 278	. 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 304	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066   ⊆ wss 3891  ∪ cuni 4844  ran crn 5589  ‘cfv 6430  (class class class)co 7268  ℝ^*cxr 10992  ℝ⁺crp 12712  topGenctg 17129  ∞Metcxmet 20563  ballcbl 20565  MetOpencmopn 20568  TopBasesctb 22076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932  ax-pre-sup 10933

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-er 8472  df-map 8591  df-en 8708  df-dom 8709  df-sdom 8710  df-sup 9162  df-inf 9163  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-div 11616  df-nn 11957  df-2 12019  df-n0 12217  df-z 12303  df-uz 12565  df-q 12671  df-rp 12713  df-xneg 12830  df-xadd 12831  df-xmul 12832  df-topgen 17135  df-psmet 20570  df-xmet 20571  df-bl 20573  df-mopn 20574  df-bases 22077

This theorem is referenced by:  metequiv  23646  metss2  23649