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| Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 26845. (Contributed by David Moews, 28-Feb-2017.) | 
| Ref | Expression | 
|---|---|
| ang.1 | ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) | 
| angvald.1 | ⊢ (𝜑 → 𝑋 ∈ ℂ) | 
| angvald.2 | ⊢ (𝜑 → 𝑋 ≠ 0) | 
| angvald.3 | ⊢ (𝜑 → 𝑌 ∈ ℂ) | 
| angvald.4 | ⊢ (𝜑 → 𝑌 ≠ 0) | 
| Ref | Expression | 
|---|---|
| angvald | ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | angvald.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℂ) | |
| 2 | angvald.2 | . 2 ⊢ (𝜑 → 𝑋 ≠ 0) | |
| 3 | angvald.3 | . 2 ⊢ (𝜑 → 𝑌 ∈ ℂ) | |
| 4 | angvald.4 | . 2 ⊢ (𝜑 → 𝑌 ≠ 0) | |
| 5 | ang.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) | |
| 6 | 5 | angval 26845 | . 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) | 
| 7 | 1, 2, 3, 4, 6 | syl22anc 838 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 {csn 4625 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ℂcc 11154 0cc0 11156 / cdiv 11921 ℑcim 15138 logclog 26597 | 
| 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-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 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-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 | 
| This theorem is referenced by: angcld 26849 angrteqvd 26850 cosangneg2d 26851 ang180lem4 26856 lawcos 26860 isosctrlem3 26864 angpieqvdlem2 26873 | 
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