Theorem readdcli 8092

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 8058	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 + 𝐵) ∈ ℝ

Syntax hints:   ∈ wcel 2177  (class class class)co 5951  ℝcr 7931   + caddc 7935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addrcl 8029

This theorem is referenced by:  resubcli  8342  eqneg  8812  2re  9113  3re  9117  4re  9120  5re  9122  6re  9124  7re  9126  8re  9128  9re  9130  numltc  9536  ef01bndlem  12111  ex-fl  15735