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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 → A B = A B

Proof

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