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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	$⊢ A \in ℂ$
	pncan3i.2	$⊢ B \in ℂ$
Assertion	negsubi	$⊢ A + (- B) = A - B$

Proof

Step	Hyp	Ref	Expression
1		negidi.1	$⊢ A \in ℂ$
2		pncan3i.2	$⊢ B \in ℂ$
3		negsub	$⊢ (A \in ℂ \land B \in ℂ) \to A + (- B) = A - B$
4	1 2 3	mp2an	$⊢ A + (- B) = A - B$