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Theorem bdmopn 15234

Description: The standard bounded metric corresponding to 𝐶 generates the same topology as 𝐶. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)

**Hypotheses**
Ref	Expression
stdbdmet.1	⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^*, < ))
stdbdmopn.2	⊢ 𝐽 = (MetOpen‘𝐶)

**Assertion**
Ref	Expression
bdmopn	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))

Distinct variable groups: 𝑥,𝑦,𝐶 𝑥,𝑅,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐷(𝑥,𝑦) 𝐽(𝑥,𝑦)

Proof of Theorem bdmopn Dummy variables 𝑟 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpxr 9896	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
2	1	ad2antll 491	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ^*)
3		simpl2 1027	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
4		xrmincl 11831	. . . . . . 7 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑟, 𝑅}, ℝ^, < ) ∈ ℝ^*)
5	2, 3, 4	syl2anc 411	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^, < ) ∈ ℝ^)
6		rpre 9895	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
7	6	ad2antll 491	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ)
8		0xr 8226	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
9	8	a1i 9	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ∈ ℝ^*)
10		rpgt0 9900	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
11	10	ad2antll 491	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑟)
12		simpl3 1028	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑅)
13		xrltmininf 11835	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑟, 𝑅}, ℝ^, < ) ↔ (0 < 𝑟 ∧ 0 < 𝑅)))
14	8, 2, 3, 13	mp3an2i 1378	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (0 < inf({𝑟, 𝑅}, ℝ^*, < ) ↔ (0 < 𝑟 ∧ 0 < 𝑅)))
15	11, 12, 14	mpbir2and 952	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < inf({𝑟, 𝑅}, ℝ^*, < ))
16	9, 5, 15	xrltled 10034	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ≤ inf({𝑟, 𝑅}, ℝ^*, < ))
17		xrmin1inf 11832	. . . . . . 7 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟)
18	2, 3, 17	syl2anc 411	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ≤ 𝑟)
19		xrrege0 10060	. . . . . 6 ⊢ (((inf({𝑟, 𝑅}, ℝ^, < ) ∈ ℝ^ ∧ 𝑟 ∈ ℝ) ∧ (0 ≤ inf({𝑟, 𝑅}, ℝ^, < ) ∧ inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟)) → inf({𝑟, 𝑅}, ℝ^*, < ) ∈ ℝ)
20	5, 7, 16, 18, 19	syl22anc 1274	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ∈ ℝ)
21	20, 15	elrpd 9928	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ∈ ℝ⁺)
22		simprl 531	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑧 ∈ 𝑋)
23		xrmin2inf 11833	. . . . . . . 8 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑅)
24	2, 3, 23	syl2anc 411	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ≤ 𝑅)
25	22, 5, 24	3jca 1203	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑧 ∈ 𝑋 ∧ inf({𝑟, 𝑅}, ℝ^, < ) ∈ ℝ^ ∧ inf({𝑟, 𝑅}, ℝ^*, < ) ≤ 𝑅))
26		stdbdmet.1	. . . . . . 7 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^*, < ))
27	26	bdbl 15233	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ inf({𝑟, 𝑅}, ℝ^, < ) ∈ ℝ^ ∧ inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑅)) → (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )) = (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^*, < )))
28	25, 27	syldan 282	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )) = (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )))
29	28	eqcomd 2237	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )))
30		breq1 4091	. . . . . 6 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑠 ≤ 𝑟 ↔ inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟))
31		oveq2 6026	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )))
32		oveq2 6026	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐷)𝑠) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )))
33	31, 32	eqeq12d 2246	. . . . . 6 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → ((𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠) ↔ (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^*, < ))))
34	30, 33	anbi12d 473	. . . . 5 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → ((𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)) ↔ (inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟 ∧ (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )))))
35	34	rspcev 2910	. . . 4 ⊢ ((inf({𝑟, 𝑅}, ℝ^, < ) ∈ ℝ⁺ ∧ (inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟 ∧ (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )))) → ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)))
36	21, 18, 29, 35	syl12anc 1271	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)))
37	36	ralrimivva 2614	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ∀𝑧 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)))
38		simp1 1023	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
39	26	bdxmet 15231	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
40		stdbdmopn.2	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
41		eqid 2231	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
42	40, 41	metequiv2 15226	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑧 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)) → 𝐽 = (MetOpen‘𝐷)))
43	38, 39, 42	syl2anc 411	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → (∀𝑧 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)) → 𝐽 = (MetOpen‘𝐷)))
44	37, 43	mpd 13	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 ∀wral 2510 ∃wrex 2511 {cpr 3670 class class class wbr 4088 ‘cfv 5326 (class class class)co 6018 ∈ cmpo 6020 infcinf 7182 ℝcr 8031 0cc0 8032 ℝ^*cxr 8213 < clt 8214 ≤ cle 8215 ℝ⁺crp 9888 ∞Metcxmet 14556 ballcbl 14558 MetOpencmopn 14561

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-map 6819 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-icc 10130 df-seqfrec 10711 df-exp 10802 df-cj 11407 df-re 11408 df-im 11409 df-rsqrt 11563 df-abs 11564 df-topgen 13348 df-psmet 14563 df-xmet 14564 df-bl 14566 df-mopn 14567 df-top 14728 df-bases 14773

This theorem is referenced by: mopnex 15235

W3C validator