Theorem readdcli 8339

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 8305	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	mp2an 430	1 ⊢ (𝐴 + 𝐵) ∈ ℝ

This proof depends on syntax axioms:   ∈ wcel 2209  (class class class)co 6085  ℝcr 8178   + caddc 8182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addrcl 8276

This theorem is used by:  resubcli  8589  eqneg  9063  2re  9375  3re  9379  4re  9382  5re  9384  6re  9386  7re  9388  8re  9390  9re  9392  numltc  9804  ef01bndlem  12525  ballotfilem2  13230  log2ublog2  16092  ex-fl  16751