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Mirrors > Home > MPE Home > Th. List > cnbl0 | Structured version Visualization version GIF version |
Description: Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.) |
Ref | Expression |
---|---|
cnblcld.1 | ⊢ 𝐷 = (abs ∘ − ) |
Ref | Expression |
---|---|
cnbl0 | ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1090 | . . . . . 6 ⊢ (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) < 𝑅) ↔ (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)) ∧ (abs‘𝑥) < 𝑅)) | |
2 | abscl 15220 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ) | |
3 | absge0 15229 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 0 ≤ (abs‘𝑥)) | |
4 | 2, 3 | jca 513 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥))) |
5 | 4 | adantl 483 | . . . . . . 7 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥))) |
6 | 5 | biantrurd 534 | . . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) < 𝑅 ↔ (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥)) ∧ (abs‘𝑥) < 𝑅))) |
7 | 1, 6 | bitr4id 290 | . . . . 5 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → (((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) < 𝑅) ↔ (abs‘𝑥) < 𝑅)) |
8 | 0re 11211 | . . . . . 6 ⊢ 0 ∈ ℝ | |
9 | simpl 484 | . . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → 𝑅 ∈ ℝ*) | |
10 | elico2 13383 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑥) ∈ (0[,)𝑅) ↔ ((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) < 𝑅))) | |
11 | 8, 9, 10 | sylancr 588 | . . . . 5 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ (0[,)𝑅) ↔ ((abs‘𝑥) ∈ ℝ ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) < 𝑅))) |
12 | 0cn 11201 | . . . . . . . . 9 ⊢ 0 ∈ ℂ | |
13 | cnblcld.1 | . . . . . . . . . . 11 ⊢ 𝐷 = (abs ∘ − ) | |
14 | 13 | cnmetdval 24268 | . . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘(0 − 𝑥))) |
15 | abssub 15268 | . . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(0 − 𝑥)) = (abs‘(𝑥 − 0))) | |
16 | 14, 15 | eqtrd 2773 | . . . . . . . . 9 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘(𝑥 − 0))) |
17 | 12, 16 | mpan 689 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0𝐷𝑥) = (abs‘(𝑥 − 0))) |
18 | subid1 11475 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
19 | 18 | fveq2d 6891 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (abs‘(𝑥 − 0)) = (abs‘𝑥)) |
20 | 17, 19 | eqtrd 2773 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (0𝐷𝑥) = (abs‘𝑥)) |
21 | 20 | adantl 483 | . . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘𝑥)) |
22 | 21 | breq1d 5156 | . . . . 5 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((0𝐷𝑥) < 𝑅 ↔ (abs‘𝑥) < 𝑅)) |
23 | 7, 11, 22 | 3bitr4d 311 | . . . 4 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ (0[,)𝑅) ↔ (0𝐷𝑥) < 𝑅)) |
24 | 23 | pm5.32da 580 | . . 3 ⊢ (𝑅 ∈ ℝ* → ((𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,)𝑅)) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) < 𝑅))) |
25 | absf 15279 | . . . . 5 ⊢ abs:ℂ⟶ℝ | |
26 | ffn 6713 | . . . . 5 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
27 | 25, 26 | ax-mp 5 | . . . 4 ⊢ abs Fn ℂ |
28 | elpreima 7054 | . . . 4 ⊢ (abs Fn ℂ → (𝑥 ∈ (◡abs “ (0[,)𝑅)) ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,)𝑅)))) | |
29 | 27, 28 | mp1i 13 | . . 3 ⊢ (𝑅 ∈ ℝ* → (𝑥 ∈ (◡abs “ (0[,)𝑅)) ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,)𝑅)))) |
30 | cnxmet 24270 | . . . . 5 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
31 | 13, 30 | eqeltri 2830 | . . . 4 ⊢ 𝐷 ∈ (∞Met‘ℂ) |
32 | elbl 23875 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ (0(ball‘𝐷)𝑅) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) < 𝑅))) | |
33 | 31, 12, 32 | mp3an12 1452 | . . 3 ⊢ (𝑅 ∈ ℝ* → (𝑥 ∈ (0(ball‘𝐷)𝑅) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) < 𝑅))) |
34 | 24, 29, 33 | 3bitr4d 311 | . 2 ⊢ (𝑅 ∈ ℝ* → (𝑥 ∈ (◡abs “ (0[,)𝑅)) ↔ 𝑥 ∈ (0(ball‘𝐷)𝑅))) |
35 | 34 | eqrdv 2731 | 1 ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5146 ◡ccnv 5673 “ cima 5677 ∘ ccom 5678 Fn wfn 6534 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 ℂcc 11103 ℝcr 11104 0cc0 11105 ℝ*cxr 11242 < clt 11243 ≤ cle 11244 − cmin 11439 [,)cico 13321 abscabs 15176 ∞Metcxmet 20913 ballcbl 20915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-n0 12468 df-z 12554 df-uz 12818 df-rp 12970 df-xadd 13088 df-ico 13325 df-seq 13962 df-exp 14023 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-psmet 20920 df-xmet 20921 df-met 20922 df-bl 20923 |
This theorem is referenced by: psercnlem2 25917 efopnlem1 26145 binomcxplemdvbinom 43044 binomcxplemnotnn0 43047 |
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