Axiom ax-pre-ltadd 7865

Description: Ordering property of addition on reals. Axiom for real and complex numbers, justified by Theorem axpre-ltadd 7823. (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
ax-pre-ltadd	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )

Detailed syntax breakdown of Axiom ax-pre-ltadd

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cr 7748	. . . 4 class RR
3	1, 2	wcel 2136	. . 3 wff A e. RR
4		cB	. . . 4 class B
5	4, 2	wcel 2136	. . 3 wff B e. RR
6		cC	. . . 4 class C
7	6, 2	wcel 2136	. . 3 wff C e. RR
8	3, 5, 7	w3a 968	. 2 wff ( A e. RR /\ B e. RR /\ C e. RR )
9		cltrr 7753	. . . 4 class <RR
10	1, 4, 9	wbr 3981	. . 3 wff A <RR B
11		caddc 7752	. . . . 5 class +
12	6, 1, 11	co 5841	. . . 4 class ( C + A )
13	6, 4, 11	co 5841	. . . 4 class ( C + B )
14	12, 13, 9	wbr 3981	. . 3 wff ( C + A ) <RR ( C + B )
15	10, 14	wi 4	. 2 wff ( A <RR B -> ( C + A ) <RR ( C + B ) )
16	8, 15	wi 4	1 wff ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )

This axiom is referenced by:  axltadd  7964