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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 ( abs ‘ ( 𝐴 − 𝐵 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) )

Proof

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