Theorem absdivd

Description: Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	abssubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	absdivd.2	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	absdivd	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		abssubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		absdivd.2	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		absdiv	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) )