![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > atandm | Structured version Visualization version GIF version |
Description: Since the property is a little lengthy, we abbreviate 𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i as 𝐴 ∈ dom arctan. This is the necessary precondition for the definition of arctan to make sense. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
atandm | ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3550 | . . 3 ⊢ (𝐴 ∈ (ℂ ∖ {-i, i}) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ {-i, i})) | |
2 | elprg 4144 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ {-i, i} ↔ (𝐴 = -i ∨ 𝐴 = i))) | |
3 | 2 | notbid 307 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (¬ 𝐴 ∈ {-i, i} ↔ ¬ (𝐴 = -i ∨ 𝐴 = i))) |
4 | neanior 2874 | . . . . 5 ⊢ ((𝐴 ≠ -i ∧ 𝐴 ≠ i) ↔ ¬ (𝐴 = -i ∨ 𝐴 = i)) | |
5 | 3, 4 | syl6bbr 277 | . . . 4 ⊢ (𝐴 ∈ ℂ → (¬ 𝐴 ∈ {-i, i} ↔ (𝐴 ≠ -i ∧ 𝐴 ≠ i))) |
6 | 5 | pm5.32i 667 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ {-i, i}) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ≠ -i ∧ 𝐴 ≠ i))) |
7 | 1, 6 | bitri 263 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ {-i, i}) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ≠ -i ∧ 𝐴 ≠ i))) |
8 | ovex 6577 | . . . 4 ⊢ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ V | |
9 | df-atan 24394 | . . . 4 ⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) | |
10 | 8, 9 | dmmpti 5936 | . . 3 ⊢ dom arctan = (ℂ ∖ {-i, i}) |
11 | 10 | eleq2i 2680 | . 2 ⊢ (𝐴 ∈ dom arctan ↔ 𝐴 ∈ (ℂ ∖ {-i, i})) |
12 | 3anass 1035 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ≠ -i ∧ 𝐴 ≠ i))) | |
13 | 7, 11, 12 | 3bitr4i 291 | 1 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {cpr 4127 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 logclog 24105 arctancatan 24391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-ov 6552 df-atan 24394 |
This theorem is referenced by: atandm2 24404 atandm3 24405 atancj 24437 2efiatan 24445 tanatan 24446 dvatan 24462 |
Copyright terms: Public domain | W3C validator |