Theorem absmuld

Description: Absolute value distributes over multiplication. Proposition 10-3.7(f) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	abssubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	absmuld	⊢ ( 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		absmul	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 · 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( abs ‘ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( abs ‘ ( 𝐴 · 𝐵 ) ) = ( ( abs ‘ 𝐴 ) · ( abs ‘ 𝐵 ) ) )