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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 25387. (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 25387 | . 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
7 | 1, 2, 3, 4, 6 | syl22anc 837 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 {csn 4525 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ℂcc 10524 0cc0 10526 / cdiv 11286 ℑcim 14449 logclog 25146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: angcld 25391 angrteqvd 25392 cosangneg2d 25393 ang180lem4 25398 lawcos 25402 isosctrlem3 25406 angpieqvdlem2 25415 |
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