Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cnblcld | Structured version Visualization version GIF version |
Description: Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.) |
Ref | Expression |
---|---|
cnblcld.1 | ⊢ 𝐷 = (abs ∘ − ) |
Ref | Expression |
---|---|
cnblcld | ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,]𝑅)) = {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absf 14866 | . . . . 5 ⊢ abs:ℂ⟶ℝ | |
2 | ffn 6523 | . . . . 5 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
3 | elpreima 6856 | . . . . 5 ⊢ (abs Fn ℂ → (𝑥 ∈ (◡abs “ (0[,]𝑅)) ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,]𝑅)))) | |
4 | 1, 2, 3 | mp2b 10 | . . . 4 ⊢ (𝑥 ∈ (◡abs “ (0[,]𝑅)) ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,]𝑅))) |
5 | df-3an 1091 | . . . . . . 7 ⊢ (((abs‘𝑥) ∈ ℝ* ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≤ 𝑅) ↔ (((abs‘𝑥) ∈ ℝ* ∧ 0 ≤ (abs‘𝑥)) ∧ (abs‘𝑥) ≤ 𝑅)) | |
6 | abscl 14807 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ) | |
7 | 6 | rexrd 10848 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ*) |
8 | absge0 14816 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 0 ≤ (abs‘𝑥)) | |
9 | 7, 8 | jca 515 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((abs‘𝑥) ∈ ℝ* ∧ 0 ≤ (abs‘𝑥))) |
10 | 9 | adantl 485 | . . . . . . . 8 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ ℝ* ∧ 0 ≤ (abs‘𝑥))) |
11 | 10 | biantrurd 536 | . . . . . . 7 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ≤ 𝑅 ↔ (((abs‘𝑥) ∈ ℝ* ∧ 0 ≤ (abs‘𝑥)) ∧ (abs‘𝑥) ≤ 𝑅))) |
12 | 5, 11 | bitr4id 293 | . . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → (((abs‘𝑥) ∈ ℝ* ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≤ 𝑅) ↔ (abs‘𝑥) ≤ 𝑅)) |
13 | 0xr 10845 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
14 | simpl 486 | . . . . . . 7 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → 𝑅 ∈ ℝ*) | |
15 | elicc1 12944 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑥) ∈ (0[,]𝑅) ↔ ((abs‘𝑥) ∈ ℝ* ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≤ 𝑅))) | |
16 | 13, 14, 15 | sylancr 590 | . . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ (0[,]𝑅) ↔ ((abs‘𝑥) ∈ ℝ* ∧ 0 ≤ (abs‘𝑥) ∧ (abs‘𝑥) ≤ 𝑅))) |
17 | 0cn 10790 | . . . . . . . . . 10 ⊢ 0 ∈ ℂ | |
18 | cnblcld.1 | . . . . . . . . . . . 12 ⊢ 𝐷 = (abs ∘ − ) | |
19 | 18 | cnmetdval 23622 | . . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘(0 − 𝑥))) |
20 | abssub 14855 | . . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(0 − 𝑥)) = (abs‘(𝑥 − 0))) | |
21 | 19, 20 | eqtrd 2771 | . . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘(𝑥 − 0))) |
22 | 17, 21 | mpan 690 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (0𝐷𝑥) = (abs‘(𝑥 − 0))) |
23 | subid1 11063 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
24 | 23 | fveq2d 6699 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (abs‘(𝑥 − 0)) = (abs‘𝑥)) |
25 | 22, 24 | eqtrd 2771 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0𝐷𝑥) = (abs‘𝑥)) |
26 | 25 | adantl 485 | . . . . . . 7 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → (0𝐷𝑥) = (abs‘𝑥)) |
27 | 26 | breq1d 5049 | . . . . . 6 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((0𝐷𝑥) ≤ 𝑅 ↔ (abs‘𝑥) ≤ 𝑅)) |
28 | 12, 16, 27 | 3bitr4d 314 | . . . . 5 ⊢ ((𝑅 ∈ ℝ* ∧ 𝑥 ∈ ℂ) → ((abs‘𝑥) ∈ (0[,]𝑅) ↔ (0𝐷𝑥) ≤ 𝑅)) |
29 | 28 | pm5.32da 582 | . . . 4 ⊢ (𝑅 ∈ ℝ* → ((𝑥 ∈ ℂ ∧ (abs‘𝑥) ∈ (0[,]𝑅)) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) ≤ 𝑅))) |
30 | 4, 29 | syl5bb 286 | . . 3 ⊢ (𝑅 ∈ ℝ* → (𝑥 ∈ (◡abs “ (0[,]𝑅)) ↔ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) ≤ 𝑅))) |
31 | 30 | abbi2dv 2867 | . 2 ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,]𝑅)) = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) ≤ 𝑅)}) |
32 | df-rab 3060 | . 2 ⊢ {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅} = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (0𝐷𝑥) ≤ 𝑅)} | |
33 | 31, 32 | eqtr4di 2789 | 1 ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,]𝑅)) = {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 {cab 2714 {crab 3055 class class class wbr 5039 ◡ccnv 5535 “ cima 5539 ∘ ccom 5540 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 ℝ*cxr 10831 ≤ cle 10833 − cmin 11027 [,]cicc 12903 abscabs 14762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-icc 12907 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |