Theorem readdcli 8175

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 8141	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 + 𝐵) ∈ ℝ

Syntax hints:   ∈ wcel 2200  (class class class)co 6010  ℝcr 8014   + caddc 8018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addrcl 8112

This theorem is referenced by:  resubcli  8425  eqneg  8895  2re  9196  3re  9200  4re  9203  5re  9205  6re  9207  7re  9209  8re  9211  9re  9213  numltc  9619  ef01bndlem  12288  ex-fl  16198