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Metamath Proof Explorer

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 ) )