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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 3919 | . . 3 ⊢ (𝐴 ∈ (ℂ ∖ {-i, i}) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ {-i, i})) | |
2 | elprg 4606 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ {-i, i} ↔ (𝐴 = -i ∨ 𝐴 = i))) | |
3 | 2 | notbid 318 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (¬ 𝐴 ∈ {-i, i} ↔ ¬ (𝐴 = -i ∨ 𝐴 = i))) |
4 | neanior 3036 | . . . . 5 ⊢ ((𝐴 ≠ -i ∧ 𝐴 ≠ i) ↔ ¬ (𝐴 = -i ∨ 𝐴 = i)) | |
5 | 3, 4 | bitr4di 289 | . . . 4 ⊢ (𝐴 ∈ ℂ → (¬ 𝐴 ∈ {-i, i} ↔ (𝐴 ≠ -i ∧ 𝐴 ≠ i))) |
6 | 5 | pm5.32i 576 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ {-i, i}) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ≠ -i ∧ 𝐴 ≠ i))) |
7 | 1, 6 | bitri 275 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ {-i, i}) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ≠ -i ∧ 𝐴 ≠ i))) |
8 | ovex 7385 | . . . 4 ⊢ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))) ∈ V | |
9 | df-atan 26169 | . . . 4 ⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) | |
10 | 8, 9 | dmmpti 6643 | . . 3 ⊢ dom arctan = (ℂ ∖ {-i, i}) |
11 | 10 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ dom arctan ↔ 𝐴 ∈ (ℂ ∖ {-i, i})) |
12 | 3anass 1096 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ≠ -i ∧ 𝐴 ≠ i))) | |
13 | 7, 11, 12 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∖ cdif 3906 {cpr 4587 dom cdm 5632 ‘cfv 6494 (class class class)co 7352 ℂcc 11008 1c1 11011 ici 11012 + caddc 11013 · cmul 11015 − cmin 11344 -cneg 11345 / cdiv 11771 2c2 12167 logclog 25862 arctancatan 26166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6446 df-fun 6496 df-fn 6497 df-fv 6502 df-ov 7355 df-atan 26169 |
This theorem is referenced by: atandm2 26179 atandm3 26180 atancj 26212 2efiatan 26220 tanatan 26221 dvatan 26237 |
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