Theorem 2ebits

Description: The bits of a power of two. (Contributed by Mario Carneiro, 5-Sep-2016)

	Ref	Expression
Assertion	2ebits	$$⊢N\in {\mathbb{N}}_0\to \mathrm{bits}\left(2^N\right)=\left\{N\right\}$$

Proof

Step	Hyp	Ref	Expression
1		2nn	$$⊢2\in \mathbb{N}$$
2	1	a1i	$$⊢N\in {\mathbb{N}}_0\to 2\in \mathbb{N}$$
3		id	$$⊢N\in {\mathbb{N}}_0\to N\in {\mathbb{N}}_0$$
4	2 3	nnexpcld	$$⊢N\in {\mathbb{N}}_0\to 2^N\in \mathbb{N}$$
5	4	nncnd	$$⊢N\in {\mathbb{N}}_0\to 2^N\in ℂ$$
6		oveq2	$$⊢k=N\to 2^k=2^N$$
7	6	sumsn	$$⊢\left(N\in {\mathbb{N}}_0\wedge 2^N\in ℂ\right)\to \sum _{k\in \left\{N\right\}}{2}^k=2^N$$
8	5 7	mpdan	$$⊢N\in {\mathbb{N}}_0\to \sum _{k\in \left\{N\right\}}{2}^k=2^N$$
9	8	fveq2d	$$⊢N\in {\mathbb{N}}_0\to \mathrm{bits}\left(\sum _{k\in \left\{N\right\}}{2}^k\right)=\mathrm{bits}\left(2^N\right)$$
10		snssi	$$⊢N\in {\mathbb{N}}_0\to \left\{N\right\}\subseteq {\mathbb{N}}_0$$
11		snfi	$$⊢\left\{N\right\}\in \mathrm{Fin}$$
12		elfpw	$$⊢\left\{N\right\}\in \left(𝒫{\mathbb{N}}_0\cap \mathrm{Fin}\right)\leftrightarrow \left(\left\{N\right\}\subseteq {\mathbb{N}}_0\wedge \left\{N\right\}\in \mathrm{Fin}\right)$$
13	10 11 12	sylanblrc	$$⊢N\in {\mathbb{N}}_0\to \left\{N\right\}\in \left(𝒫{\mathbb{N}}_0\cap \mathrm{Fin}\right)$$
14		bitsinv2	$$⊢\left\{N\right\}\in \left(𝒫{\mathbb{N}}_0\cap \mathrm{Fin}\right)\to \mathrm{bits}\left(\sum _{k\in \left\{N\right\}}{2}^k\right)=\left\{N\right\}$$
15	13 14	syl	$$⊢N\in {\mathbb{N}}_0\to \mathrm{bits}\left(\sum _{k\in \left\{N\right\}}{2}^k\right)=\left\{N\right\}$$
16	9 15	eqtr3d	$$⊢N\in {\mathbb{N}}_0\to \mathrm{bits}\left(2^N\right)=\left\{N\right\}$$