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Theorem atans 26974
Description: The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.)
Hypotheses
Ref Expression
atansopn.d 𝐷 = (ℂ ∖ (-∞(,]0))
atansopn.s 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
Assertion
Ref Expression
atans (𝐴𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐷
Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem atans
StepHypRef Expression
1 oveq1 7439 . . . 4 (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2))
21oveq2d 7448 . . 3 (𝑦 = 𝐴 → (1 + (𝑦↑2)) = (1 + (𝐴↑2)))
32eleq1d 2825 . 2 (𝑦 = 𝐴 → ((1 + (𝑦↑2)) ∈ 𝐷 ↔ (1 + (𝐴↑2)) ∈ 𝐷))
4 atansopn.s . 2 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
53, 4elrab2 3694 1 (𝐴𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  {crab 3435  cdif 3947  (class class class)co 7432  cc 11154  0cc0 11156  1c1 11157   + caddc 11159  -∞cmnf 11294  2c2 12322  (,]cioc 13389  cexp 14103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  atans2  26975
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