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Mirrors > Home > MPE Home > Th. List > mulcld | Structured version Visualization version GIF version |
Description: Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
mulcld | ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | addcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mulcl 10939 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ) |
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