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Metamath Proof Explorer

Theorem expp1d

Description: Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of Gleason p. 134. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	$⊢ φ \to A \in ℂ$
	expcld.2	$⊢ φ \to N \in ℕ_{0}$
Assertion	expp1d	$⊢ φ \to A^{N + 1} = A^{N} A$

Proof

Step	Hyp	Ref	Expression
1		expcld.1	$⊢ φ \to A \in ℂ$
2		expcld.2	$⊢ φ \to N \in ℕ_{0}$
3		expp1	$⊢ (A \in ℂ \land N \in ℕ_{0}) \to A^{N + 1} = A^{N} A$
4	1 2 3	syl2anc	$⊢ φ \to A^{N + 1} = A^{N} A$