Theorem decbin0

Description: Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014)

	Ref	Expression
Hypothesis	decbin.1	⊢ 𝐴 ∈ ℕ0
Assertion	decbin0	⊢ ( 4 · 𝐴 ) = ( 2 · ( 2 · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		decbin.1	⊢ 𝐴 ∈ ℕ0
2		2t2e4	⊢ ( 2 · 2 ) = 4
3	2	oveq1i	⊢ ( ( 2 · 2 ) · 𝐴 ) = ( 4 · 𝐴 )
4		2cn	⊢ 2 ∈ ℂ
5	1	nn0cni	⊢ 𝐴 ∈ ℂ
6	4 4 5	mulassi	⊢ ( ( 2 · 2 ) · 𝐴 ) = ( 2 · ( 2 · 𝐴 ) )
7	3 6	eqtr3i	⊢ ( 4 · 𝐴 ) = ( 2 · ( 2 · 𝐴 ) )