Theorem absmuli

Description: Absolute value distributes over multiplication. Proposition 10-3.7(f) of Gleason p. 133. (Contributed by NM, 1-Oct-1999)

	Ref	Expression
Hypotheses	absvalsqi.1	⊢ 𝐴 ∈ ℂ
	abssub.2	⊢ 𝐵 ∈ ℂ
Assertion	absmuli	⊢ ( abs ‘ ( 𝐴 · 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( abs ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		absvalsqi.1	⊢ 𝐴 ∈ ℂ
2		abssub.2	⊢ 𝐵 ∈ ℂ
3		absmul	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 · 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( abs ‘ 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( abs ‘ ( 𝐴 · 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( abs ‘ 𝐵 ) )