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Theorem stdbdmopn 24406

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

**Hypotheses**
Ref	Expression
stdbdmet.1	⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
stdbdmopn.2	⊢ 𝐽 = (MetOpen‘𝐶)

**Assertion**
Ref	Expression
stdbdmopn	⊢ ((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 stdbdmopn Dummy variables 𝑟 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpxr 12961	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
2	1	ad2antll 729	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ^*)
3		simpl2 1193	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
4	2, 3	ifcld 4535	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ^*)
5		rpre 12960	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
6	5	ad2antll 729	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ)
7		rpgt0 12964	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
8	7	ad2antll 729	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑟)
9		simpl3 1194	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑅)
10		breq2 5111	. . . . . . . . 9 ⊢ (𝑟 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (0 < 𝑟 ↔ 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
11		breq2 5111	. . . . . . . . 9 ⊢ (𝑅 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (0 < 𝑅 ↔ 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
12	10, 11	ifboth 4528	. . . . . . . 8 ⊢ ((0 < 𝑟 ∧ 0 < 𝑅) → 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅))
13	8, 9, 12	syl2anc 584	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅))
14		0xr 11221	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
15		xrltle 13109	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ^*) → (0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → 0 ≤ if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
16	14, 4, 15	sylancr 587	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → 0 ≤ if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
17	13, 16	mpd 15	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 ≤ if(𝑟 ≤ 𝑅, 𝑟, 𝑅))
18		xrmin1 13137	. . . . . . 7 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟)
19	2, 3, 18	syl2anc 584	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟)
20		xrrege0 13134	. . . . . 6 ⊢ (((if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ^* ∧ 𝑟 ∈ ℝ) ∧ (0 ≤ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∧ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ)
21	4, 6, 17, 19, 20	syl22anc 838	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ)
22	21, 13	elrpd 12992	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ⁺)
23		simprl 770	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑧 ∈ 𝑋)
24		xrmin2 13138	. . . . . . . 8 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑅)
25	2, 3, 24	syl2anc 584	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑅)
26	23, 4, 25	3jca 1128	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑧 ∈ 𝑋 ∧ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ^* ∧ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑅))
27		stdbdmet.1	. . . . . . 7 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
28	27	stdbdbl 24405	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ^* ∧ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑅)) → (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)) = (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
29	26, 28	syldan 591	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)) = (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
30	29	eqcomd 2735	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)) = (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
31		breq1 5110	. . . . . 6 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑠 ≤ 𝑟 ↔ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟))
32		oveq2 7395	. . . . . . 7 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
33		oveq2 7395	. . . . . . 7 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑧(ball‘𝐷)𝑠) = (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
34	32, 33	eqeq12d 2745	. . . . . 6 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → ((𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠) ↔ (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)) = (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅))))
35	31, 34	anbi12d 632	. . . . 5 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → ((𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)) ↔ (if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟 ∧ (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)) = (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))))
36	35	rspcev 3588	. . . 4 ⊢ ((if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ⁺ ∧ (if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟 ∧ (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)) = (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))) → ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)))
37	22, 19, 30, 36	syl12anc 836	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)))
38	37	ralrimivva 3180	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → ∀𝑧 ∈ 𝑋 ∀𝑟 ∈ ℝ⁺ ∃𝑠 ∈ ℝ⁺ (𝑠 ≤ 𝑟 ∧ (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐷)𝑠)))
39		simp1 1136	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
40	27	stdbdxmet 24403	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
41		stdbdmopn.2	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
42		eqid 2729	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
43	41, 42	metequiv2 24398	. . 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)))
44	39, 40, 43	syl2anc 584	. 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)))
45	38, 44	mpd 15	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ifcif 4488 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 ℝcr 11067 0cc0 11068 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 ℝ⁺crp 12951 ∞Metcxmet 21249 ballcbl 21251 MetOpencmopn 21254

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-icc 13313 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-bl 21259 df-mopn 21260 df-bases 22833

This theorem is referenced by: mopnex 24407 xlebnum 24864

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