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Metamath Proof Explorer

Theorem resubcli

Description: Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses renegcl.1 ⊢ 𝐴 ∈ ℝ
resubcl.2 ⊢ 𝐵 ∈ ℝ
Assertion resubcli ( 𝐴 − 𝐵 ) ∈ ℝ

Proof

Step Hyp Ref Expression
1 renegcl.1 ⊢ 𝐴 ∈ ℝ
2 resubcl.2 ⊢ 𝐵 ∈ ℝ
3 1 recni ⊢ 𝐴 ∈ ℂ
4 2 recni ⊢ 𝐵 ∈ ℂ
5 negsub ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) )
6 3 4 5 mp2an ⊢ ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 )
7 2 renegcli ⊢ - 𝐵 ∈ ℝ
8 1 7 readdcli ⊢ ( 𝐴 + - 𝐵 ) ∈ ℝ
9 6 8 eqeltrri ⊢ ( 𝐴 − 𝐵 ) ∈ ℝ