Theorem negsubd

Description: Relationship between subtraction and negative. Theorem I.3 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	negsubd	⊢ ( 𝜑 → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )