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Theorem stdbdmopn 24474

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 12927	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
2	1	ad2antll 730	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ^*)
3		simpl2 1194	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
4	2, 3	ifcld 4528	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ^*)
5		rpre 12926	. . . . . . 7 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
6	5	ad2antll 730	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ)
7		rpgt0 12930	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
8	7	ad2antll 730	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑟)
9		simpl3 1195	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑅)
10		breq2 5104	. . . . . . . . 9 ⊢ (𝑟 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (0 < 𝑟 ↔ 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
11		breq2 5104	. . . . . . . . 9 ⊢ (𝑅 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (0 < 𝑅 ↔ 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
12	10, 11	ifboth 4521	. . . . . . . 8 ⊢ ((0 < 𝑟 ∧ 0 < 𝑅) → 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅))
13	8, 9, 12	syl2anc 585	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅))
14		0xr 11191	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
15		xrltle 13075	. . . . . . . 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 588	. . . . . . 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 13104	. . . . . . 7 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟)
19	2, 3, 18	syl2anc 585	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟)
20		xrrege0 13101	. . . . . 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 839	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ)
22	21, 13	elrpd 12958	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ⁺)
23		simprl 771	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → 𝑧 ∈ 𝑋)
24		xrmin2 13105	. . . . . . . 8 ⊢ ((𝑟 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑅)
25	2, 3, 24	syl2anc 585	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑅)
26	23, 4, 25	3jca 1129	. . . . . 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 24473	. . . . . 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 592	. . . . 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 2743	. . . 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 5103	. . . . . 6 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑠 ≤ 𝑟 ↔ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟))
32		oveq2 7376	. . . . . . 7 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
33		oveq2 7376	. . . . . . 7 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑧(ball‘𝐷)𝑠) = (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
34	32, 33	eqeq12d 2753	. . . . . 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 633	. . . . 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 3578	. . . 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 837	. . 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 3181	. 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 1137	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐶 ∈ (∞Met‘𝑋))
40	27	stdbdxmet 24471	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
41		stdbdmopn.2	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
42		eqid 2737	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
43	41, 42	metequiv2 24466	. . 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 585	. 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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ifcif 4481 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 ℝcr 11037 0cc0 11038 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 ℝ⁺crp 12917 ∞Metcxmet 21306 ballcbl 21308 MetOpencmopn 21311

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-icc 13280 df-topgen 17375 df-psmet 21313 df-xmet 21314 df-bl 21316 df-mopn 21317 df-bases 22902

This theorem is referenced by: mopnex 24475 xlebnum 24932

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