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Mirrors > Home > MPE Home > Th. List > atans | Structured version Visualization version GIF version |
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 7166 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2)) | |
2 | 1 | oveq2d 7175 | . . 3 ⊢ (𝑦 = 𝐴 → (1 + (𝑦↑2)) = (1 + (𝐴↑2))) |
3 | 2 | eleq1d 2900 | . 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 3686 | 1 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {crab 3145 ∖ cdif 3936 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 -∞cmnf 10676 2c2 11695 (,]cioc 12742 ↑cexp 13432 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: atans2 25512 |
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