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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 24331. (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 24331 | . 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
7 | 1, 2, 3, 4, 6 | syl22anc 1319 | 1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℂcc 9813 0cc0 9815 / cdiv 10563 ℑcim 13686 logclog 24105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: angcld 24335 angrteqvd 24336 cosangneg2d 24337 ang180lem4 24342 lawcos 24346 isosctrlem3 24350 angpieqvdlem2 24356 |
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