Theorem binom21

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

	Ref	Expression
Assertion	binom21	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 1 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · 𝐴 ) ) + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		binom2	⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐴 + 1 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 1 ) ) ) + ( 1 ↑ 2 ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 1 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 1 ) ) ) + ( 1 ↑ 2 ) ) )
4		mulid1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · 1 ) = 𝐴 )
5	4	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( 2 · ( 𝐴 · 1 ) ) = ( 2 · 𝐴 ) )
6	5	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 1 ) ) ) = ( ( 𝐴 ↑ 2 ) + ( 2 · 𝐴 ) ) )
7		sq1	⊢ ( 1 ↑ 2 ) = 1
8	7	a1i	⊢ ( 𝐴 ∈ ℂ → ( 1 ↑ 2 ) = 1 )
9	6 8	oveq12d	⊢ ( 𝐴 ∈ ℂ → ( ( ( 𝐴 ↑ 2 ) + ( 2 · ( 𝐴 · 1 ) ) ) + ( 1 ↑ 2 ) ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · 𝐴 ) ) + 1 ) )
10	3 9	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 1 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) + ( 2 · 𝐴 ) ) + 1 ) )