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Mirrors > Home > MPE Home > Th. List > bndatandm | Structured version Visualization version GIF version |
Description: A point in the open unit disk is in the domain of the arctangent. (Contributed by Mario Carneiro, 5-Apr-2015.) |
Ref | Expression |
---|---|
bndatandm | ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 475 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ) | |
2 | sqcl 13179 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
3 | 2 | adantr 473 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑2) ∈ ℂ) |
4 | 3 | abscld 14516 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) ∈ ℝ) |
5 | 2nn0 11599 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
6 | absexp 14385 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) | |
7 | 1, 5, 6 | sylancl 581 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) |
8 | simpr 478 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1) | |
9 | abscl 14359 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
10 | 9 | adantr 473 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ) |
11 | 1red 10329 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℝ) | |
12 | absge0 14368 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
13 | 12 | adantr 473 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ≤ (abs‘𝐴)) |
14 | 0le1 10843 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ≤ 1) |
16 | 10, 11, 13, 15 | lt2sqd 13299 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘𝐴) < 1 ↔ ((abs‘𝐴)↑2) < (1↑2))) |
17 | 8, 16 | mpbid 224 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘𝐴)↑2) < (1↑2)) |
18 | sq1 13212 | . . . . . 6 ⊢ (1↑2) = 1 | |
19 | 17, 18 | syl6breq 4884 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘𝐴)↑2) < 1) |
20 | 7, 19 | eqbrtrd 4865 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) < 1) |
21 | 4, 20 | ltned 10463 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) ≠ 1) |
22 | fveq2 6411 | . . . . 5 ⊢ ((𝐴↑2) = -1 → (abs‘(𝐴↑2)) = (abs‘-1)) | |
23 | ax-1cn 10282 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
24 | 23 | absnegi 14480 | . . . . . 6 ⊢ (abs‘-1) = (abs‘1) |
25 | abs1 14378 | . . . . . 6 ⊢ (abs‘1) = 1 | |
26 | 24, 25 | eqtri 2821 | . . . . 5 ⊢ (abs‘-1) = 1 |
27 | 22, 26 | syl6eq 2849 | . . . 4 ⊢ ((𝐴↑2) = -1 → (abs‘(𝐴↑2)) = 1) |
28 | 27 | necon3i 3003 | . . 3 ⊢ ((abs‘(𝐴↑2)) ≠ 1 → (𝐴↑2) ≠ -1) |
29 | 21, 28 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑2) ≠ -1) |
30 | atandm3 24957 | . 2 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | |
31 | 1, 29, 30 | sylanbrc 579 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 class class class wbr 4843 dom cdm 5312 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 ℝcr 10223 0cc0 10224 1c1 10225 < clt 10363 ≤ cle 10364 -cneg 10557 2c2 11368 ℕ0cn0 11580 ↑cexp 13114 abscabs 14315 arctancatan 24943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-atan 24946 |
This theorem is referenced by: atantayl 25016 log2cnv 25023 |
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