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Theorem metequiv2 24340

Description: If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)

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

**Assertion**
Ref	Expression
metequiv2	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → 𝐽 = 𝐾))

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 metequiv2**
Step	Hyp	Ref	Expression
1		simprrr 779	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠))
2		simplll 772	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → 𝐶 ∈ (∞Met‘𝑋))
3		simplr 766	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → 𝑥 ∈ 𝑋)
4		simprlr 777	. . . . . . . . . . . . 13 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → 𝑠 ∈ ℝ⁺)
5	4	rpxrd 13013	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → 𝑠 ∈ ℝ^*)
6		simprll 776	. . . . . . . . . . . . 13 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → 𝑟 ∈ ℝ⁺)
7	6	rpxrd 13013	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → 𝑟 ∈ ℝ^*)
8		simprrl 778	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → 𝑠 ≤ 𝑟)
9		ssbl 24250	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑠 ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) ∧ 𝑠 ≤ 𝑟) → (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟))
10	2, 3, 5, 7, 8, 9	syl221anc 1378	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟))
11	1, 10	eqsstrrd 4013	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟))
12		simpllr 773	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → 𝐷 ∈ (∞Met‘𝑋))
13		ssbl 24250	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑠 ∈ ℝ^* ∧ 𝑟 ∈ ℝ^*) ∧ 𝑠 ≤ 𝑟) → (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))
14	12, 3, 5, 7, 8, 13	syl221anc 1378	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))
15	1, 14	eqsstrd 4012	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))
16	11, 15	jca 511	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)))) → ((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟)))
17	16	expr 456	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑟 ∈ ℝ⁺ ∧ 𝑠 ∈ ℝ⁺)) → ((𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → ((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))))
18	17	anassrs 467	. . . . . . 7 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑠 ∈ ℝ⁺) → ((𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → ((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))))
19	18	reximdva 3160	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → ∃𝑠 ∈ ℝ⁺ ((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))))
20		r19.40 3111	. . . . . 6 ⊢ (∃𝑠 ∈ ℝ⁺ ((𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟)) → (∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟)))
21	19, 20	syl6 35	. . . . 5 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → (∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))))
22	21	ralimdva 3159	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → ∀𝑟 ∈ ℝ⁺ (∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))))
23		r19.26 3103	. . . 4 ⊢ (∀𝑟 ∈ ℝ⁺ (∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟)) ↔ (∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟)))
24	22, 23	imbitrdi 250	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → (∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))))
25	24	ralimdva 3159	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → ∀𝑥 ∈ 𝑋 (∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))))
26		metequiv.3	. . 3 ⊢ 𝐽 = (MetOpen‘𝐶)
27		metequiv.4	. . 3 ⊢ 𝐾 = (MetOpen‘𝐷)
28	26, 27	metequiv 24339	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 = 𝐾 ↔ ∀𝑥 ∈ 𝑋 (∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑥(ball‘𝐶)𝑠) ⊆ (𝑥(ball‘𝐷)𝑟))))
29	25, 28	sylibrd 259	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → 𝐽 = 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 ⊆ wss 3940 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 ℝ^*cxr 11243 ≤ cle 11245 ℝ⁺crp 12970 ∞Metcxmet 21212 ballcbl 21214 MetOpencmopn 21217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-topgen 17387 df-psmet 21219 df-xmet 21220 df-bl 21222 df-mopn 21223 df-bases 22770

This theorem is referenced by: stdbdmopn 24348

W3C validator