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Mirrors > Home > MPE Home > Th. List > absmuld | Structured version Visualization version GIF version |
Description: Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
abssubd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
absmuld | ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | abssubd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | absmul 13882 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵))) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵))) |
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