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Theorem atans 24457
 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 6556 . . . 4 (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2))
21oveq2d 6565 . . 3 (𝑦 = 𝐴 → (1 + (𝑦↑2)) = (1 + (𝐴↑2)))
32eleq1d 2672 . 2 (𝑦 = 𝐴 → ((1 + (𝑦↑2)) ∈ 𝐷 ↔ (1 + (𝐴↑2)) ∈ 𝐷))
4 atansopn.s . 2 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
53, 4elrab2 3333 1 (𝐴𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818  -∞cmnf 9951  2c2 10947  (,]cioc 12047  ↑cexp 12722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  atans2  24458
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