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Theorem bdmopn 12673

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 9449	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
2	1	ad2antll 482	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ^*)
3		simpl2 985	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
4		xrmincl 11035	. . . . . . 7 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑟, 𝑅}, ℝ^, < ) ∈ ℝ^*)
5	2, 3, 4	syl2anc 408	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^, < ) ∈ ℝ^)
6		rpre 9448	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
7	6	ad2antll 482	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ)
8		0xr 7812	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
9	8	a1i 9	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ∈ ℝ^*)
10		rpgt0 9453	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
11	10	ad2antll 482	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑟)
12		simpl3 986	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑅)
13		xrltmininf 11039	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑟, 𝑅}, ℝ^, < ) ↔ (0 < 𝑟 ∧ 0 < 𝑅)))
14	8, 2, 3, 13	mp3an2i 1320	. . . . . . . 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 928	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < inf({𝑟, 𝑅}, ℝ^*, < ))
16	9, 5, 15	xrltled 9585	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ≤ inf({𝑟, 𝑅}, ℝ^*, < ))
17		xrmin1inf 11036	. . . . . . 7 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟)
18	2, 3, 17	syl2anc 408	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ≤ 𝑟)
19		xrrege0 9608	. . . . . 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 1217	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ∈ ℝ)
21	20, 15	elrpd 9481	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ∈ ℝ⁺)
22		simprl 520	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑧 ∈ 𝑋)
23		xrmin2inf 11037	. . . . . . . 8 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑅)
24	2, 3, 23	syl2anc 408	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ≤ 𝑅)
25	22, 5, 24	3jca 1161	. . . . . 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 12672	. . . . . 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 280	. . . . 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 2145	. . . 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 3932	. . . . . 6 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑠 ≤ 𝑟 ↔ inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟))
31		oveq2 5782	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )))
32		oveq2 5782	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐷)𝑠) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )))
33	31, 32	eqeq12d 2154	. . . . . 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 464	. . . . 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 2789	. . . 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 1214	. . 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 2514	. 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 981	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
39	26	bdxmet 12670	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
40		stdbdmopn.2	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
41		eqid 2139	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
42	40, 41	metequiv2 12665	. . 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 408	. 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 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 {cpr 3528 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 infcinf 6870 ℝcr 7619 0cc0 7620 ℝ^*cxr 7799 < clt 7800 ≤ cle 7801 ℝ⁺crp 9441 ∞Metcxmet 12149 ballcbl 12151 MetOpencmopn 12154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-xneg 9559 df-xadd 9560 df-icc 9678 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-topgen 12141 df-psmet 12156 df-xmet 12157 df-bl 12159 df-mopn 12160 df-top 12165 df-bases 12210

This theorem is referenced by: mopnex 12674

W3C validator