This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem absdivzi

Description: Absolute value distributes over division. (Contributed by NM, 26-Mar-2005)

Ref Expression
Hypotheses absvalsqi.1 ⊢ 𝐴 ∈ ℂ
abssub.2 ⊢ 𝐵 ∈ ℂ
Assertion absdivzi ( 𝐵 ≠ 0 → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 absvalsqi.1 ⊢ 𝐴 ∈ ℂ
2 abssub.2 ⊢ 𝐵 ∈ ℂ
3 absdiv ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) )
4 1 2 3 mp3an12 ⊢ ( 𝐵 ≠ 0 → ( abs ‘ ( 𝐴 / 𝐵 ) ) = ( ( abs ‘ 𝐴 ) / ( abs ‘ 𝐵 ) ) )