Theorem metss 15010

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 14958	. . . 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 14958	. . . 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 3225	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ (topGen‘ran (ball‘𝐶)) ⊆ (topGen‘ran (ball‘𝐷))))
8		blbas 14949	. . . 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 14939	. . . . 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 14939	. . . . 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 2242	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → ∪ ran (ball‘𝐶) = ∪ ran (ball‘𝐷))
15		tgss2 14595	. . 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 2709	. . 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 14946	. . . . . . . 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 2507	. . . . 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 9790	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
23		blelrn 14936	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
24	22, 23	syl3an3 1285	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘𝐶)𝑟) ∈ ran (ball‘𝐶))
25		blcntr 14932	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)𝑟))
26		eleq2 2270	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑥(ball‘𝐶)𝑟)))
27		sseq2 3218	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥(ball‘𝐶)𝑟) → ((𝑥(ball‘𝐷)𝑠) ⊆ 𝑦 ↔ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
28	27	rexbidv 2508	. . . . . . . . . . . . 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 2874	. . . . . . . . . . 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 1206	. . . . . . . 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 2587	. . . . . 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 14944	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ran (ball‘𝐶) ∧ 𝑥 ∈ 𝑦) → ∃𝑟 ∈ ℝ+ (𝑥(ball‘𝐶)𝑟) ⊆ 𝑦)
37	36	3expb 1207	. . . . . . . . . . . 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 2644	. . . . . . . . . . . 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 3202	. . . . . . . . . . . . . . . 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 2608	. . . . . . . . . . . . . 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 2620	. . . . . . . . . . . 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 2587	. . . . . 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 2503	. . 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 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486   ⊆ wss 3167  ∪ cuni 3852  ran crn 4680  ‘cfv 5276  (class class class)co 5951  ℝ^*cxr 8113  ℝ⁺crp 9782  topGenctg 13130  ∞Metcxmet 14342  ballcbl 14344  MetOpencmopn 14347  TopBasesctb 14558

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-map 6744  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-xneg 9901  df-xadd 9902  df-seqfrec 10600  df-exp 10691  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-topgen 13136  df-psmet 14349  df-xmet 14350  df-bl 14352  df-mopn 14353  df-top 14514  df-bases 14559

This theorem is referenced by:  metequiv  15011  metss2  15014