Theorem atans 26846

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 7396	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2))
2	1	oveq2d 7405	. . 3 ⊢ (𝑦 = 𝐴 → (1 + (𝑦↑2)) = (1 + (𝐴↑2)))
3	2	eleq1d 2814	. 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 3664	1 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408   ∖ cdif 3913  (class class class)co 7389  ℂcc 11072  0cc0 11074  1c1 11075   + caddc 11077  -∞cmnf 11212  2c2 12242  (,]cioc 13313  ↑cexp 14032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-iota 6466  df-fv 6521  df-ov 7392

This theorem is referenced by:  atans2  26847