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Axiom ax-pre-ltadd 9868
Description: Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 9844. Normally new proofs would use axltadd 9962. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-ltadd ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Detailed syntax breakdown of Axiom ax-pre-ltadd
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 9791 . . . 4 class
31, 2wcel 1976 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1976 . . 3 wff 𝐵 ∈ ℝ
6 cC . . . 4 class 𝐶
76, 2wcel 1976 . . 3 wff 𝐶 ∈ ℝ
83, 5, 7w3a 1030 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)
9 cltrr 9796 . . . 4 class <
101, 4, 9wbr 4577 . . 3 wff 𝐴 < 𝐵
11 caddc 9795 . . . . 5 class +
126, 1, 11co 6526 . . . 4 class (𝐶 + 𝐴)
136, 4, 11co 6526 . . . 4 class (𝐶 + 𝐵)
1412, 13, 9wbr 4577 . . 3 wff (𝐶 + 𝐴) < (𝐶 + 𝐵)
1510, 14wi 4 . 2 wff (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵))
168, 15wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵)))
Colors of variables: wff setvar class
This axiom is referenced by:  axltadd  9962
  Copyright terms: Public domain W3C validator