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Metamath Proof Explorer

Theorem absdivd

Description: Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses abscld.1 ⊢ φ → A ∈ ℂ
abssubd.2 ⊢ φ → B ∈ ℂ
absdivd.2 ⊢ φ → B ≠ 0
Assertion absdivd ⊢ φ → A B = A B

Proof

Step Hyp Ref Expression
1 abscld.1 ⊢ φ → A ∈ ℂ
2 abssubd.2 ⊢ φ → B ∈ ℂ
3 absdivd.2 ⊢ φ → B ≠ 0
4 absdiv ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ B ≠ 0 → A B = A B
5 1 2 3 4 syl3anc ⊢ φ → A B = A B