![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > angvald | Structured version Visualization version GIF version |
Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 26859. (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 26859 | . 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
7 | 1, 2, 3, 4, 6 | syl22anc 839 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11151 0cc0 11153 / cdiv 11918 ℑcim 15134 logclog 26611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: angcld 26863 angrteqvd 26864 cosangneg2d 26865 ang180lem4 26870 lawcos 26874 isosctrlem3 26878 angpieqvdlem2 26887 |
Copyright terms: Public domain | W3C validator |