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	⊢ (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 7439	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2))
2	1	oveq2d 7448	. . 3 ⊢ (𝑦 = 𝐴 → (1 + (𝑦↑2)) = (1 + (𝐴↑2)))
3	2	eleq1d 2825	. 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 3694	1 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {crab 3435   ∖ cdif 3947  (class class class)co 7432  ℂcc 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