Theorem abssub

Description: Swapping order of subtraction doesn't change the absolute value. (Contributed by NM, 1-Oct-1999) (Proof shortened by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	abssub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		subcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 − 𝐵 ) ∈ ℂ )
2		absneg	⊢ ( ( 𝐴 − 𝐵 ) ∈ ℂ → ( abs ‘ - ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐴 − 𝐵 ) ) )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ - ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐴 − 𝐵 ) ) )
4		negsubdi2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 − 𝐵 ) = ( 𝐵 − 𝐴 ) )
5	4	fveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ - ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )
6	3 5	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) )