Theorem binom21

Description: Special case of binom2 where B = 1 . (Contributed by Scott Fenton, 11-May-2014)

	Ref	Expression
Assertion	binom21	$$⊢A\in ℂ\to {\left(A+1\right)}^2=A^2+2A+1$$

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	$$⊢1\in ℂ$$
2		binom2	$$⊢\left(A\in ℂ\wedge 1\in ℂ\right)\to {\left(A+1\right)}^2=A^2+2A\cdot 1+1^2$$
3	1 2	mpan2	$$⊢A\in ℂ\to {\left(A+1\right)}^2=A^2+2A\cdot 1+1^2$$
4		mulid1	$$⊢A\in ℂ\to A\cdot 1=A$$
5	4	oveq2d	$$⊢A\in ℂ\to 2A\cdot 1=2A$$
6	5	oveq2d	$$⊢A\in ℂ\to A^2+2A\cdot 1=A^2+2A$$
7		sq1	$$⊢1^2=1$$
8	7	a1i	$$⊢A\in ℂ\to 1^2=1$$
9	6 8	oveq12d	$$⊢A\in ℂ\to A^2+2A\cdot 1+1^2=A^2+2A+1$$
10	3 9	eqtrd	$$⊢A\in ℂ\to {\left(A+1\right)}^2=A^2+2A+1$$