Theorem atans 26988

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 7438	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2))
2	1	oveq2d 7447	. . 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 3698	1 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {crab 3433   ∖ cdif 3960  (class class class)co 7431  ℂcc 11151  0cc0 11153  1c1 11154   + caddc 11156  -∞cmnf 11291  2c2 12319  (,]cioc 13385  ↑cexp 14099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  atans2  26989