Theorem readdcli 8292

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 8258	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 + 𝐵) ∈ ℝ

Syntax hints:   ∈ wcel 2205  (class class class)co 6052  ℝcr 8131   + caddc 8135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-addrcl 8229

This theorem is referenced by:  resubcli  8541  eqneg  9011  2re  9312  3re  9316  4re  9319  5re  9321  6re  9323  7re  9325  8re  9327  9re  9329  numltc  9740  ef01bndlem  12450  ballotfilem2  13153  ex-fl  16542