Metamath Proof Explorer

Theorem nnpw2pb

Description: A number is a positive integer iff it can be represented as the sum of a power of 2 and a "remainder" smaller than the power. (Contributed by AV, 31-May-2020)

		Ref	Expression
	Assertion	nnpw2pb	$⊢ N \in ℕ \leftrightarrow \exists i \in ℕ_{0} \exists r \in (0 ..^2^{i}) N = 2^{i} + r$

Proof

Step	Hyp	Ref	Expression
1		nnpw2p	$⊢ N \in ℕ \to \exists i \in ℕ_{0} \exists r \in (0 ..^2^{i}) N = 2^{i} + r$
2		2nn	$⊢ 2 \in ℕ$
3		nnexpcl	$⊢ (2 \in ℕ \land i \in ℕ_{0}) \to 2^{i} \in ℕ$
4	2 3	mpan	$⊢ i \in ℕ_{0} \to 2^{i} \in ℕ$
5		elfzonn0	$⊢ r \in (0 ..^2^{i}) \to r \in ℕ_{0}$
6		nnnn0addcl	$⊢ (2^{i} \in ℕ \land r \in ℕ_{0}) \to 2^{i} + r \in ℕ$
7	4 5 6	syl2an	$⊢ (i \in ℕ_{0} \land r \in (0 ..^2^{i})) \to 2^{i} + r \in ℕ$
8		eleq1	$⊢ N = 2^{i} + r \to (N \in ℕ \leftrightarrow 2^{i} + r \in ℕ)$
9	7 8	syl5ibrcom	$⊢ (i \in ℕ_{0} \land r \in (0 ..^2^{i})) \to (N = 2^{i} + r \to N \in ℕ)$
10	9	rexlimivv	$⊢ \exists i \in ℕ_{0} \exists r \in (0 ..^2^{i}) N = 2^{i} + r \to N \in ℕ$
11	1 10	impbii	$⊢ N \in ℕ \leftrightarrow \exists i \in ℕ_{0} \exists r \in (0 ..^2^{i}) N = 2^{i} + r$