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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