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Metamath Proof Explorer

Theorem negsubi

Description: Relationship between subtraction and negative. Theorem I.3 of Apostol p. 18. (Contributed by NM, 26-Nov-1994) (Proof shortened by Andrew Salmon, 22-Oct-2011)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
Assertion	negsubi	⊢ ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 )