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Metamath Proof Explorer

Theorem 1p1times

Description: Two times a number. (Contributed by NM, 18-May-1999) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion 1p1times ( 𝐴 ∈ ℂ → ( ( 1 + 1 ) · 𝐴 ) = ( 𝐴 + 𝐴 ) )

Proof

Step Hyp Ref Expression
1 1cnd ⊢ ( 𝐴 ∈ ℂ → 1 ∈ ℂ )
2 id ⊢ ( 𝐴 ∈ ℂ → 𝐴 ∈ ℂ )
3 mullid ⊢ ( 𝐴 ∈ ℂ → ( 1 · 𝐴 ) = 𝐴 )
4 3 3 oveq12d ⊢ ( 𝐴 ∈ ℂ → ( ( 1 · 𝐴 ) + ( 1 · 𝐴 ) ) = ( 𝐴 + 𝐴 ) )
5 1 2 1 4 joinlmuladdmuld ⊢ ( 𝐴 ∈ ℂ → ( ( 1 + 1 ) · 𝐴 ) = ( 𝐴 + 𝐴 ) )