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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 476 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ) | |
2 | sqcl 13219 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
3 | 2 | adantr 474 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑2) ∈ ℂ) |
4 | 3 | abscld 14552 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) ∈ ℝ) |
5 | 2nn0 11637 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
6 | absexp 14421 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) | |
7 | 1, 5, 6 | sylancl 580 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) |
8 | simpr 479 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1) | |
9 | abscl 14395 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
10 | 9 | adantr 474 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ) |
11 | 1red 10357 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℝ) | |
12 | absge0 14404 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
13 | 12 | adantr 474 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ≤ (abs‘𝐴)) |
14 | 0le1 10875 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ≤ 1) |
16 | 10, 11, 13, 15 | lt2sqd 13339 | . . . . . . 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 13252 | . . . . . 6 ⊢ (1↑2) = 1 | |
19 | 17, 18 | syl6breq 4914 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘𝐴)↑2) < 1) |
20 | 7, 19 | eqbrtrd 4895 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) < 1) |
21 | 4, 20 | ltned 10492 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) ≠ 1) |
22 | fveq2 6433 | . . . . 5 ⊢ ((𝐴↑2) = -1 → (abs‘(𝐴↑2)) = (abs‘-1)) | |
23 | ax-1cn 10310 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
24 | 23 | absnegi 14516 | . . . . . 6 ⊢ (abs‘-1) = (abs‘1) |
25 | abs1 14414 | . . . . . 6 ⊢ (abs‘1) = 1 | |
26 | 24, 25 | eqtri 2849 | . . . . 5 ⊢ (abs‘-1) = 1 |
27 | 22, 26 | syl6eq 2877 | . . . 4 ⊢ ((𝐴↑2) = -1 → (abs‘(𝐴↑2)) = 1) |
28 | 27 | necon3i 3031 | . . 3 ⊢ ((abs‘(𝐴↑2)) ≠ 1 → (𝐴↑2) ≠ -1) |
29 | 21, 28 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑2) ≠ -1) |
30 | atandm3 25018 | . 2 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | |
31 | 1, 29, 30 | sylanbrc 578 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 class class class wbr 4873 dom cdm 5342 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 1c1 10253 < clt 10391 ≤ cle 10392 -cneg 10586 2c2 11406 ℕ0cn0 11618 ↑cexp 13154 abscabs 14351 arctancatan 25004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-atan 25007 |
This theorem is referenced by: atantayl 25077 log2cnv 25084 |
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