Axiom ax-pre-ltadd 10853

Description: Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by Theorem axpre-ltadd 10829. Normally new proofs would use axltadd 10954. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
ax-pre-ltadd	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))

Detailed syntax breakdown of Axiom ax-pre-ltadd

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cr 10776	. . . 4 class ℝ
3	1, 2	wcel 2112	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2112	. . 3 wff 𝐵 ∈ ℝ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2112	. . 3 wff 𝐶 ∈ ℝ
8	3, 5, 7	w3a 1089	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9		cltrr 10781	. . . 4 class <ℝ
10	1, 4, 9	wbr 5070	. . 3 wff 𝐴 <ℝ 𝐵
11		caddc 10780	. . . . 5 class +
12	6, 1, 11	co 7252	. . . 4 class (𝐶 + 𝐴)
13	6, 4, 11	co 7252	. . . 4 class (𝐶 + 𝐵)
14	12, 13, 9	wbr 5070	. . 3 wff (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)
15	10, 14	wi 4	. 2 wff (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))
16	8, 15	wi 4	1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵)))

This axiom is referenced by:  axltadd  10954