Theorem sqval

Description: Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004)

	Ref	Expression
Assertion	sqval	$$⊢A\in ℂ\to A^2=AA$$

Proof

Step	Hyp	Ref	Expression
1		df-2	$$⊢2=1+1$$
2	1	oveq2i	$$⊢A^2=A^{1+1}$$
3		1nn0	$$⊢1\in {\mathbb{N}}_0$$
4		expp1	$$⊢\left(A\in ℂ\wedge 1\in {\mathbb{N}}_0\right)\to A^{1+1}=A^{1}A$$
5	3 4	mpan2	$$⊢A\in ℂ\to A^{1+1}=A^{1}A$$
6	2 5	eqtrid	$$⊢A\in ℂ\to A^2=A^{1}A$$
7		exp1	$$⊢A\in ℂ\to A^1=A$$
8	7	oveq1d	$$⊢A\in ℂ\to A^{1}A=AA$$
9	6 8	eqtrd	$$⊢A\in ℂ\to A^2=AA$$