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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 25373. (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 25373 | . 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
7 | 1, 2, 3, 4, 6 | syl22anc 836 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 {csn 4560 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ℂcc 10529 0cc0 10531 / cdiv 11291 ℑcim 14451 logclog 25132 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 |
This theorem is referenced by: angcld 25377 angrteqvd 25378 cosangneg2d 25379 ang180lem4 25384 lawcos 25388 isosctrlem3 25392 angpieqvdlem2 25401 |
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