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Metamath Proof Explorer

Theorem abssubi

Description: Swapping order of subtraction doesn't change the absolute value. Example of Apostol p. 363. (Contributed by NM, 1-Oct-1999)

Ref Expression
Hypotheses absvalsqi.1 ⊢ A ∈ ℂ
abssub.2 ⊢ B ∈ ℂ
Assertion abssubi ⊢ A − B = B − A

Proof

Step Hyp Ref Expression
1 absvalsqi.1 ⊢ A ∈ ℂ
2 abssub.2 ⊢ B ∈ ℂ
3 abssub ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B = B − A
4 1 2 3 mp2an ⊢ A − B = B − A