Theorem readdcli 7912

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 7879	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	mp2an 423	1 ⊢ (𝐴 + 𝐵) ∈ ℝ

Syntax hints:   ∈ wcel 2136  (class class class)co 5842  ℝcr 7752   + caddc 7756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 107  ax-addrcl 7850

This theorem is referenced by:  resubcli  8161  eqneg  8628  2re  8927  3re  8931  4re  8934  5re  8936  6re  8938  7re  8940  8re  8942  9re  8944  numltc  9347  ef01bndlem  11697  ex-fl  13606