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Theorem atans 25507
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 7147 . . . 4 (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2))
21oveq2d 7156 . . 3 (𝑦 = 𝐴 → (1 + (𝑦↑2)) = (1 + (𝐴↑2)))
32eleq1d 2900 . 2 (𝑦 = 𝐴 → ((1 + (𝑦↑2)) ∈ 𝐷 ↔ (1 + (𝐴↑2)) ∈ 𝐷))
4 atansopn.s . 2 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
53, 4elrab2 3668 1 (𝐴𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  {crab 3136  cdif 3915  (class class class)co 7140  cc 10522  0cc0 10524  1c1 10525   + caddc 10527  -∞cmnf 10660  2c2 11680  (,]cioc 12727  cexp 13425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-iota 6297  df-fv 6346  df-ov 7143
This theorem is referenced by:  atans2  25508
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