Theorem atans 26089

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	⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐷

Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem atans

Step	Hyp	Ref	Expression
1		oveq1 7291	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2))
2	1	oveq2d 7300	. . 3 ⊢ (𝑦 = 𝐴 → (1 + (𝑦↑2)) = (1 + (𝐴↑2)))
3	2	eleq1d 2824	. 2 ⊢ (𝑦 = 𝐴 → ((1 + (𝑦↑2)) ∈ 𝐷 ↔ (1 + (𝐴↑2)) ∈ 𝐷))
4		atansopn.s	. 2 ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷}
5	3, 4	elrab2 3628	1 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2107  {crab 3069   ∖ cdif 3885  (class class class)co 7284  ℂcc 10878  0cc0 10880  1c1 10881   + caddc 10883  -∞cmnf 11016  2c2 12037  (,]cioc 13089  ↑cexp 13791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2069  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-iota 6395  df-fv 6445  df-ov 7287

This theorem is referenced by:  atans2  26090