Theorem decbin2

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

	Ref	Expression
Hypothesis	decbin.1	|- A e. NN0
Assertion	decbin2	|- ( ( 4 x. A ) + 2 ) = ( 2 x. ( ( 2 x. A ) + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		decbin.1	|- A e. NN0
2		2t1e2	|- ( 2 x. 1 ) = 2
3	2	oveq2i	|- ( ( 2 x. ( 2 x. A ) ) + ( 2 x. 1 ) ) = ( ( 2 x. ( 2 x. A ) ) + 2 )
4		2cn	|- 2 e. CC
5	1	nn0cni	|- A e. CC
6	4 5	mulcli	|- ( 2 x. A ) e. CC
7		ax-1cn	|- 1 e. CC
8	4 6 7	adddii	|- ( 2 x. ( ( 2 x. A ) + 1 ) ) = ( ( 2 x. ( 2 x. A ) ) + ( 2 x. 1 ) )
9	1	decbin0	|- ( 4 x. A ) = ( 2 x. ( 2 x. A ) )
10	9	oveq1i	|- ( ( 4 x. A ) + 2 ) = ( ( 2 x. ( 2 x. A ) ) + 2 )
11	3 8 10	3eqtr4ri	|- ( ( 4 x. A ) + 2 ) = ( 2 x. ( ( 2 x. A ) + 1 ) )