angvald - Metamath Proof Explorer

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Theorem angvald 26729

Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 26726. (Contributed by David Moews, 28-Feb-2017.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
angvald	⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))

Distinct variable groups: 𝑥,𝑦,𝑋 𝑥,𝑌,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐹(𝑥,𝑦)

**Proof of Theorem angvald**
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 26726	. 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
7	1, 2, 3, 4, 6	syl22anc 838	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∖ cdif 3942 {csn 4624 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ℂcc 11130 0cc0 11132 / cdiv 11895 ℑcim 15071 logclog 26481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419

This theorem is referenced by: angcld 26730 angrteqvd 26731 cosangneg2d 26732 ang180lem4 26737 lawcos 26741 isosctrlem3 26745 angpieqvdlem2 26754

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