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Theorem stdbdmopn 24462

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 12915	. . . . . . . 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 4526	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ∈ ℝ^*)
5		rpre 12914	. . . . . . 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 12918	. . . . . . . . 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 5102	. . . . . . . . 9 ⊢ (𝑟 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (0 < 𝑟 ↔ 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
11		breq2 5102	. . . . . . . . 9 ⊢ (𝑅 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (0 < 𝑅 ↔ 0 < if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
12	10, 11	ifboth 4519	. . . . . . . 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 11179	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
15		xrltle 13063	. . . . . . . 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 13092	. . . . . . 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 13089	. . . . . 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 12946	. . . 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 13093	. . . . . . . 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 24461	. . . . . 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 2742	. . . 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 5101	. . . . . 6 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑠 ≤ 𝑟 ↔ if(𝑟 ≤ 𝑅, 𝑟, 𝑅) ≤ 𝑟))
32		oveq2 7366	. . . . . . 7 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑧(ball‘𝐶)𝑠) = (𝑧(ball‘𝐶)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
33		oveq2 7366	. . . . . . 7 ⊢ (𝑠 = if(𝑟 ≤ 𝑅, 𝑟, 𝑅) → (𝑧(ball‘𝐷)𝑠) = (𝑧(ball‘𝐷)if(𝑟 ≤ 𝑅, 𝑟, 𝑅)))
34	32, 33	eqeq12d 2752	. . . . . 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 3576	. . . 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 3179	. 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 24459	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
41		stdbdmopn.2	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐶)
42		eqid 2736	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
43	41, 42	metequiv2 24454	. . 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 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 ifcif 4479 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ℝcr 11025 0cc0 11026 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 ℝ⁺crp 12905 ∞Metcxmet 21294 ballcbl 21296 MetOpencmopn 21299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-icc 13268 df-topgen 17363 df-psmet 21301 df-xmet 21302 df-bl 21304 df-mopn 21305 df-bases 22890

This theorem is referenced by: mopnex 24463 xlebnum 24920

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