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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 7144 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦↑2) = (𝐴↑2)) | |
2 | 1 | oveq2d 7153 | . . 3 ⊢ (𝑦 = 𝐴 → (1 + (𝑦↑2)) = (1 + (𝐴↑2))) |
3 | 2 | eleq1d 2895 | . 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 3669 | 1 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3137 ∖ cdif 3916 (class class class)co 7137 ℂcc 10516 0cc0 10518 1c1 10519 + caddc 10521 -∞cmnf 10654 2c2 11674 (,]cioc 12721 ↑cexp 13414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-rab 3142 df-v 3483 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-br 5048 df-iota 6295 df-fv 6344 df-ov 7140 |
This theorem is referenced by: atans2 25490 |
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