bdmopn - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > bdmopn		GIF version

Theorem bdmopn 14918

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 9782	. . . . . . . 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 11519	. . . . . . 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 9781	. . . . . . 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 8118	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
9	8	a1i 9	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ∈ ℝ^*)
10		rpgt0 9786	. . . . . . . . 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 11523	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (0 < inf({𝑟, 𝑅}, ℝ^, < ) ↔ (0 < 𝑟 ∧ 0 < 𝑅)))
14	8, 2, 3, 13	mp3an2i 1354	. . . . . . . 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 9920	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ≤ inf({𝑟, 𝑅}, ℝ^*, < ))
17		xrmin1inf 11520	. . . . . . 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 9946	. . . . . 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 9814	. . . 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 11521	. . . . . . . 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 14917	. . . . . 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 2210	. . . 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 4046	. . . . . 6 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑠 ≤ 𝑟 ↔ inf({𝑟, 𝑅}, ℝ^, < ) ≤ 𝑟))
31		oveq2 5951	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐶)inf({𝑟, 𝑅}, ℝ^, < )))
32		oveq2 5951	. . . . . . 7 ⊢ (𝑠 = inf({𝑟, 𝑅}, ℝ^, < ) → (𝑧(ball‘𝐷)𝑠) = (𝑧(ball‘𝐷)inf({𝑟, 𝑅}, ℝ^, < )))
33	31, 32	eqeq12d 2219	. . . . . 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 2876	. . . 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 2587	. 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 14915	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
40		stdbdmopn.2	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
41		eqid 2204	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
42	40, 41	metequiv2 14910	. . 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 1372 ∈ wcel 2175 ∀wral 2483 ∃wrex 2484 {cpr 3633 class class class wbr 4043 ‘cfv 5270 (class class class)co 5943 ∈ cmpo 5945 infcinf 7084 ℝcr 7923 0cc0 7924 ℝ^*cxr 8105 < clt 8106 ≤ cle 8107 ℝ⁺crp 9774 ∞Metcxmet 14240 ballcbl 14242 MetOpencmopn 14245

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 ax-arch 8043 ax-caucvg 8044

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-isom 5279 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-frec 6476 df-map 6736 df-sup 7085 df-inf 7086 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-n0 9295 df-z 9372 df-uz 9648 df-q 9740 df-rp 9775 df-xneg 9893 df-xadd 9894 df-icc 10016 df-seqfrec 10591 df-exp 10682 df-cj 11095 df-re 11096 df-im 11097 df-rsqrt 11251 df-abs 11252 df-topgen 13034 df-psmet 14247 df-xmet 14248 df-bl 14250 df-mopn 14251 df-top 14412 df-bases 14457

This theorem is referenced by: mopnex 14919

W3C validator