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| Description: The "domain of continuity" of the arctangent. (Contributed by Mario Carneiro, 7-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| atansopn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | 
| atansopn.s | ⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} | 
| Ref | Expression | 
|---|---|
| atans | ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷)) | 
| 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)) | 
| Colors of variables: wff setvar class | 
| 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 | 
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