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Theorem bdmopn 15193

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 9869	. . . . . . . 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 1025	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
4		xrmincl 11792	. . . . . . 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 9868	. . . . . . 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 8204	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
9	8	a1i 9	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ∈ ℝ^*)
10		rpgt0 9873	. . . . . . . . 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 1026	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑅)
13		xrltmininf 11796	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑟, 𝑅}, ℝ^, < ) ↔ (0 < 𝑟 ∧ 0 < 𝑅)))
14	8, 2, 3, 13	mp3an2i 1376	. . . . . . . 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 950	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < inf({𝑟, 𝑅}, ℝ^*, < ))
16	9, 5, 15	xrltled 10007	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ≤ inf({𝑟, 𝑅}, ℝ^*, < ))
17		xrmin1inf 11793	. . . . . . 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 10033	. . . . . 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 1272	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ∈ ℝ)
21	20, 15	elrpd 9901	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ∈ ℝ⁺)
22		simprl 529	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑧 ∈ 𝑋)
23		xrmin2inf 11794	. . . . . . . 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 1201	. . . . . 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 15192	. . . . . 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 2235	. . . 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 4086	. . . . . 6 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑠 ≤ 𝑟 ↔ inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟))
31		oveq2 6015	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )))
32		oveq2 6015	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐷)𝑠) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )))
33	31, 32	eqeq12d 2244	. . . . . 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 2907	. . . 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 1269	. . 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 2612	. 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 1021	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
39	26	bdxmet 15190	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
40		stdbdmopn.2	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
41		eqid 2229	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
42	40, 41	metequiv2 15185	. . 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 1002 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 {cpr 3667 class class class wbr 4083 ‘cfv 5318 (class class class)co 6007 ∈ cmpo 6009 infcinf 7161 ℝcr 8009 0cc0 8010 ℝ^*cxr 8191 < clt 8192 ≤ cle 8193 ℝ⁺crp 9861 ∞Metcxmet 14515 ballcbl 14517 MetOpencmopn 14520

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 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

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 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-map 6805 df-sup 7162 df-inf 7163 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-xneg 9980 df-xadd 9981 df-icc 10103 df-seqfrec 10682 df-exp 10773 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525 df-topgen 13308 df-psmet 14522 df-xmet 14523 df-bl 14525 df-mopn 14526 df-top 14687 df-bases 14732

This theorem is referenced by: mopnex 15194

W3C validator