Theorem readdcli 8039

Description: Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)

Hypotheses

Ref	Expression
recni.1	⊢ 𝐴 ∈ ℝ
axri.2	⊢ 𝐵 ∈ ℝ

Assertion

Ref	Expression
readdcli	⊢ (𝐴 + 𝐵) ∈ ℝ

Proof of Theorem readdcli

Step	Hyp	Ref	Expression
1		recni.1	. 2 ⊢ 𝐴 ∈ ℝ
2		axri.2	. 2 ⊢ 𝐵 ∈ ℝ
3		readdcl 8005	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 + 𝐵) ∈ ℝ

Syntax hints:   ∈ wcel 2167  (class class class)co 5922  ℝcr 7878   + caddc 7882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addrcl 7976

This theorem is referenced by:  resubcli  8289  eqneg  8759  2re  9060  3re  9064  4re  9067  5re  9069  6re  9071  7re  9073  8re  9075  9re  9077  numltc  9482  ef01bndlem  11921  ex-fl  15371