![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 482 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ ℂ) | |
2 | sqcl 14155 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑2) ∈ ℂ) |
4 | 3 | abscld 15472 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) ∈ ℝ) |
5 | 2nn0 12541 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
6 | absexp 15340 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) | |
7 | 1, 5, 6 | sylancl 586 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) = ((abs‘𝐴)↑2)) |
8 | simpr 484 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) < 1) | |
9 | abscl 15314 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
10 | 9 | adantr 480 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘𝐴) ∈ ℝ) |
11 | 1red 11260 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 1 ∈ ℝ) | |
12 | absge0 15323 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
13 | 12 | adantr 480 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ≤ (abs‘𝐴)) |
14 | 0le1 11784 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 0 ≤ 1) |
16 | 10, 11, 13, 15 | lt2sqd 14292 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘𝐴) < 1 ↔ ((abs‘𝐴)↑2) < (1↑2))) |
17 | 8, 16 | mpbid 232 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘𝐴)↑2) < (1↑2)) |
18 | sq1 14231 | . . . . . 6 ⊢ (1↑2) = 1 | |
19 | 17, 18 | breqtrdi 5189 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → ((abs‘𝐴)↑2) < 1) |
20 | 7, 19 | eqbrtrd 5170 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) < 1) |
21 | 4, 20 | ltned 11395 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (abs‘(𝐴↑2)) ≠ 1) |
22 | fveq2 6907 | . . . . 5 ⊢ ((𝐴↑2) = -1 → (abs‘(𝐴↑2)) = (abs‘-1)) | |
23 | ax-1cn 11211 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
24 | 23 | absnegi 15436 | . . . . . 6 ⊢ (abs‘-1) = (abs‘1) |
25 | abs1 15333 | . . . . . 6 ⊢ (abs‘1) = 1 | |
26 | 24, 25 | eqtri 2763 | . . . . 5 ⊢ (abs‘-1) = 1 |
27 | 22, 26 | eqtrdi 2791 | . . . 4 ⊢ ((𝐴↑2) = -1 → (abs‘(𝐴↑2)) = 1) |
28 | 27 | necon3i 2971 | . . 3 ⊢ ((abs‘(𝐴↑2)) ≠ 1 → (𝐴↑2) ≠ -1) |
29 | 21, 28 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → (𝐴↑2) ≠ -1) |
30 | atandm3 26936 | . 2 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | |
31 | 1, 29, 30 | sylanbrc 583 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → 𝐴 ∈ dom arctan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 < clt 11293 ≤ cle 11294 -cneg 11491 2c2 12319 ℕ0cn0 12524 ↑cexp 14099 abscabs 15270 arctancatan 26922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-atan 26925 |
This theorem is referenced by: atantayl 26995 log2cnv 27002 |
Copyright terms: Public domain | W3C validator |