Theorem decbin2

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

	Ref	Expression
Hypothesis	decbin.1	⊢ 𝐴 ∈ ℕ0
Assertion	decbin2	⊢ ( ( 4 · 𝐴 ) + 2 ) = ( 2 · ( ( 2 · 𝐴 ) + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		decbin.1	⊢ 𝐴 ∈ ℕ0
2		2t1e2	⊢ ( 2 · 1 ) = 2
3	2	oveq2i	⊢ ( ( 2 · ( 2 · 𝐴 ) ) + ( 2 · 1 ) ) = ( ( 2 · ( 2 · 𝐴 ) ) + 2 )
4		2cn	⊢ 2 ∈ ℂ
5	1	nn0cni	⊢ 𝐴 ∈ ℂ
6	4 5	mulcli	⊢ ( 2 · 𝐴 ) ∈ ℂ
7		ax-1cn	⊢ 1 ∈ ℂ
8	4 6 7	adddii	⊢ ( 2 · ( ( 2 · 𝐴 ) + 1 ) ) = ( ( 2 · ( 2 · 𝐴 ) ) + ( 2 · 1 ) )
9	1	decbin0	⊢ ( 4 · 𝐴 ) = ( 2 · ( 2 · 𝐴 ) )
10	9	oveq1i	⊢ ( ( 4 · 𝐴 ) + 2 ) = ( ( 2 · ( 2 · 𝐴 ) ) + 2 )
11	3 8 10	3eqtr4ri	⊢ ( ( 4 · 𝐴 ) + 2 ) = ( 2 · ( ( 2 · 𝐴 ) + 1 ) )