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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 25381. (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 25381 | . 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 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ℂcc 10537 0cc0 10539 / cdiv 11299 ℑcim 14459 logclog 25140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 |
This theorem is referenced by: angcld 25385 angrteqvd 25386 cosangneg2d 25387 ang180lem4 25392 lawcos 25396 isosctrlem3 25400 angpieqvdlem2 25409 |
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