Theorem metss 15211

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 15159	. . . 4 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐶)))
3	2	adantr 276	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝐽 = (topGen‘ran (ball‘𝐶)))
4		metequiv.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
5	4	mopnval 15159	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (topGen‘ran (ball‘𝐷)))
6	5	adantl 277	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝐾 = (topGen‘ran (ball‘𝐷)))
7	3, 6	sseq12d 3256	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ (topGen‘ran (ball‘𝐶)) ⊆ (topGen‘ran (ball‘𝐷))))
8		blbas 15150	. . . 4 ⊢ (𝐶 ∈ (∞Met‘𝑋) → ran (ball‘𝐶) ∈ TopBases)
9	8	adantr 276	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ran (ball‘𝐶) ∈ TopBases)
10		unirnbl 15140	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐶) = 𝑋)
11	10	adantr 276	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐶) = 𝑋)
12		unirnbl 15140	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
13	12	adantl 277	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐷) = 𝑋)
14	11, 13	eqtr4d 2265	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐶) = ∪ ran (ball‘𝐷))
15		tgss2 14796	. . 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))))
16	9, 14, 15	syl2anc 411	. 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))))
17	11	raleqdv 2734	. . 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))))
18		blssex 15147	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
19	18	adantll 476	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∃𝑧 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) ↔ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
20	19	imbi2d 230	. . . . . 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)))
21	20	ralbidv 2530	. . . . 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)))
22		rpxr 9889	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
23		blelrn 15137	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
24	22, 23	syl3an3 1306	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
25		blcntr 15133	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)𝑟))
26		eleq2 2293	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑥(ball‘𝐶)𝑟)))
27		sseq2 3249	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → ((𝑥(ball‘𝐷)𝑠) ⊆ 𝑦 ↔ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
28	27	rexbidv 2531	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → (∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦 ↔ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
29	26, 28	imbi12d 234	. . . . . . . . . . . 12 ⊢ (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))))
30	29	rspcv 2904	. . . . . . . . . . 11 ⊢ ((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))))
31	30	com23 78	. . . . . . . . . 10 ⊢ ((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))))
32	24, 25, 31	sylc 62	. . . . . . . . 9 ⊢ ((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)))
33	32	3expa 1227	. . . . . . . 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)))
34	33	adantllr 481	. . . . . . 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)))
35	34	ralrimdva 2610	. . . . . 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)))
36		blss 15145	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
37	36	3expb 1228	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
38	37	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ (𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
39	38	adantlr 477	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦)) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
40		r19.29 2668	. . . . . . . . . . . 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))
41		sstr 3233	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦) → (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)
42	41	expcom 116	. . . . . . . . . . . . . . 15 ⊢ ((𝑥(ball‘𝐶)𝑟) ⊆ 𝑦 → ((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
43	42	reximdv 2631	. . . . . . . . . . . . . 14 ⊢ ((𝑥(ball‘𝐶)𝑟) ⊆ 𝑦 → (∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦))
44	43	impcom 125	. . . . . . . . . . . . 13 ⊢ ((∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦) → ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ 𝑦)
45	44	rexlimivw 2644	. . . . . . . . . . . 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)
46	40, 45	syl 14	. . . . . . . . . . 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)
47	46	ex 115	. . . . . . . . . 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))
48	39, 47	syl5com 29	. . . . . . . . 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))
49	48	expr 375	. . . . . . . 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)))
50	49	com23 78	. . . . . . 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)))
51	50	ralrimdva 2610	. . . . . 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)))
52	35, 51	impbid 129	. . . . 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)))
53	21, 52	bitrd 188	. . . 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)))
54	53	ralbidva 2526	. . 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)))
55	17, 54	bitrd 188	. 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)))
56	7, 16, 55	3bitrd 214	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   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ⊆ wss 3198  ∪ cuni 3891  ran crn 4724  ‘cfv 5324  (class class class)co 6013  ℝ^*cxr 8206  ℝ⁺crp 9881  topGenctg 13330  ∞Metcxmet 14543  ballcbl 14545  MetOpencmopn 14548  TopBasesctb 14759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-map 6814  df-sup 7177  df-inf 7178  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-xneg 10000  df-xadd 10001  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-topgen 13336  df-psmet 14550  df-xmet 14551  df-bl 14553  df-mopn 14554  df-top 14715  df-bases 14760

This theorem is referenced by:  metequiv  15212  metss2  15215