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Metamath Proof Explorer

Theorem xmulcld

Description: Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses xnegcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
xaddcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
Assertion xmulcld ( 𝜑 → ( 𝐴 ·e 𝐵 ) ∈ ℝ* )

Proof

Step Hyp Ref Expression
1 xnegcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2 xaddcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3 xmulcl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ·e 𝐵 ) ∈ ℝ* )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ·e 𝐵 ) ∈ ℝ* )