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 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 B 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 6690 . . . 4 class
31, 2wcel 1390 . . 3 wff A
4 cB . . . 4 class B
54, 2wcel 1390 . . 3 wff B
6 cC . . . 4 class 𝐶
76, 2wcel 1390 . . 3 wff 𝐶
83, 5, 7w3a 884 . 2 wff (A B 𝐶 ℝ)
9 cltrr 6695 . . . 4 class <
101, 4, 9wbr 3755 . . 3 wff A < B
11 caddc 6694 . . . . 5 class +
126, 1, 11co 5455 . . . 4 class (𝐶 + A)
136, 4, 11co 5455 . . . 4 class (𝐶 + B)
1412, 13, 9wbr 3755 . . 3 wff (𝐶 + A) < (𝐶 + B)
1510, 14wi 4 . 2 wff (A < B → (𝐶 + A) < (𝐶 + B))
168, 15wi 4 1 wff ((A B 𝐶 ℝ) → (A < B → (𝐶 + A) < (𝐶 + B)))
 Colors of variables: wff set class This axiom is referenced by:  axltadd  6866
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