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Mirrors > Home > ILE Home > Th. List > ax-pre-ltadd | GIF version |
Description: Ordering property of addition on reals. Axiom for real and complex numbers, justified by theorem axpre-ltadd 6770. (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
ax-pre-ltadd | ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → (A <ℝ B → (𝐶 + A) <ℝ (𝐶 + B))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class A | |
2 | cr 6710 | . . . 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 6715 | . . . 4 class <ℝ | |
10 | 1, 4, 9 | wbr 3755 | . . 3 wff A <ℝ B |
11 | caddc 6714 | . . . . 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))) |
Colors of variables: wff set class |
This axiom is referenced by: axltadd 6886 |
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