Theorem nn2m

Description: Multiply an element of _om by 2o . (Contributed by Scott Fenton, 16-Apr-2012) (Revised by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	nn2m	⊢ ( 𝐴 ∈ ω → ( 2o ·o 𝐴 ) = ( 𝐴 +o 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		2onn	⊢ 2o ∈ ω
2		nnmcom	⊢ ( ( 2o ∈ ω ∧ 𝐴 ∈ ω ) → ( 2o ·o 𝐴 ) = ( 𝐴 ·o 2o ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ω → ( 2o ·o 𝐴 ) = ( 𝐴 ·o 2o ) )
4		nnm2	⊢ ( 𝐴 ∈ ω → ( 𝐴 ·o 2o ) = ( 𝐴 +o 𝐴 ) )
5	3 4	eqtrd	⊢ ( 𝐴 ∈ ω → ( 2o ·o 𝐴 ) = ( 𝐴 +o 𝐴 ) )