Theorem resubcl

Description: Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997)

	Ref	Expression
Assertion	resubcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 − 𝐵 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		recn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
3		negsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
5		renegcl	⊢ ( 𝐵 ∈ ℝ → - 𝐵 ∈ ℝ )
6		readdcl	⊢ ( ( 𝐴 ∈ ℝ ∧ - 𝐵 ∈ ℝ ) → ( 𝐴 + - 𝐵 ) ∈ ℝ )
7	5 6	sylan2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + - 𝐵 ) ∈ ℝ )
8	4 7	eqeltrrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 − 𝐵 ) ∈ ℝ )