Axiom ax-pre-ltadd 6779

Description: Ordering property of addition on reals. Axiom for real and complex numbers, justified by theorem axpre-ltadd 6750. (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 6690	. . . 4 class ℝ
3	1, 2	wcel 1390	. . 3 wff A ∈ ℝ
4		cB	. . . 4 class B
5	4, 2	wcel 1390	. . 3 wff B ∈ ℝ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1390	. . 3 wff 𝐶 ∈ ℝ
8	3, 5, 7	w3a 884	. 2 wff (A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ)
9		cltrr 6695	. . . 4 class <ℝ
10	1, 4, 9	wbr 3755	. . 3 wff A <ℝ B
11		caddc 6694	. . . . 5 class +
12	6, 1, 11	co 5455	. . . 4 class (𝐶 + A)
13	6, 4, 11	co 5455	. . . 4 class (𝐶 + B)
14	12, 13, 9	wbr 3755	. . 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  6866