Theorem binom2subi

Description: Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013)

	Ref	Expression
Hypotheses	binom2subi.1	⊢ 𝐴 ∈ ℂ
	binom2subi.2	⊢ 𝐵 ∈ ℂ
Assertion	binom2subi	⊢ ( ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) − ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		binom2subi.1	⊢ 𝐴 ∈ ℂ
2		binom2subi.2	⊢ 𝐵 ∈ ℂ
3		binom2sub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) − ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) ) )
4	1 2 3	mp2an	⊢ ( ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( ( 𝐴 ↑ 2 ) − ( 2 · ( 𝐴 · 𝐵 ) ) ) + ( 𝐵 ↑ 2 ) )