Theorem sqsubswap

Description: Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013)

	Ref	Expression
Assertion	sqsubswap	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( 𝐵 − 𝐴 ) ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		subcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − 𝐵 ) ∈ ℂ )
2		sqneg	⊢ ( ( 𝐴 − 𝐵 ) ∈ ℂ → ( - ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( 𝐴 − 𝐵 ) ↑ 2 ) )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( 𝐴 − 𝐵 ) ↑ 2 ) )
4		negsubdi2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐴 ) )
5	4	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( 𝐵 − 𝐴 ) ↑ 2 ) )
6	3 5	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) ↑ 2 ) = ( ( 𝐵 − 𝐴 ) ↑ 2 ) )