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Theorem bdmopn 14683

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 9730	. . . . . . . 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 1003	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
4		xrmincl 11412	. . . . . . 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 9729	. . . . . . 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 8068	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
9	8	a1i 9	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ∈ ℝ^*)
10		rpgt0 9734	. . . . . . . . 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 1004	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑅)
13		xrltmininf 11416	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑟, 𝑅}, ℝ^, < ) ↔ (0 < 𝑟 ∧ 0 < 𝑅)))
14	8, 2, 3, 13	mp3an2i 1353	. . . . . . . 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 946	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < inf({𝑟, 𝑅}, ℝ^*, < ))
16	9, 5, 15	xrltled 9868	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ≤ inf({𝑟, 𝑅}, ℝ^*, < ))
17		xrmin1inf 11413	. . . . . . 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 9894	. . . . . 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 1250	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → inf({𝑟, 𝑅}, ℝ^*, < ) ∈ ℝ)
21	20, 15	elrpd 9762	. . . 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 11414	. . . . . . . 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 1179	. . . . . 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 14682	. . . . . 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 2199	. . . 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 4033	. . . . . 6 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑠 ≤ 𝑟 ↔ inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟))
31		oveq2 5927	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )))
32		oveq2 5927	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐷)𝑠) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )))
33	31, 32	eqeq12d 2208	. . . . . 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 2865	. . . 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 1247	. . 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 2576	. 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 999	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
39	26	bdxmet 14680	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
40		stdbdmopn.2	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
41		eqid 2193	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
42	40, 41	metequiv2 14675	. . 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 980 = wceq 1364 ∈ wcel 2164 ∀wral 2472 ∃wrex 2473 {cpr 3620 class class class wbr 4030 ‘cfv 5255 (class class class)co 5919 ∈ cmpo 5921 infcinf 7044 ℝcr 7873 0cc0 7874 ℝ^*cxr 8055 < clt 8056 ≤ cle 8057 ℝ⁺crp 9722 ∞Metcxmet 14035 ballcbl 14037 MetOpencmopn 14040

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-map 6706 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-xneg 9841 df-xadd 9842 df-icc 9964 df-seqfrec 10522 df-exp 10613 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-topgen 12874 df-psmet 14042 df-xmet 14043 df-bl 14045 df-mopn 14046 df-top 14177 df-bases 14222

This theorem is referenced by: mopnex 14684

W3C validator