MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-pre-ltadd Unicode version

Axiom ax-pre-ltadd 8813
Description: Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 8789. Normally new proofs would use axltadd 8896. (New usage is discouraged.) (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
StepHypRef Expression
1 cA . . . 4  class  A
2 cr 8736 . . . 4  class  RR
31, 2wcel 1684 . . 3  wff  A  e.  RR
4 cB . . . 4  class  B
54, 2wcel 1684 . . 3  wff  B  e.  RR
6 cC . . . 4  class  C
76, 2wcel 1684 . . 3  wff  C  e.  RR
83, 5, 7w3a 934 . 2  wff  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )
9 cltrr 8741 . . . 4  class  <RR
101, 4, 9wbr 4023 . . 3  wff  A  <RR  B
11 caddc 8740 . . . . 5  class  +
126, 1, 11co 5858 . . . 4  class  ( C  +  A )
136, 4, 11co 5858 . . . 4  class  ( C  +  B )
1412, 13, 9wbr 4023 . . 3  wff  ( C  +  A )  <RR  ( C  +  B )
1510, 14wi 4 . 2  wff  ( A 
<RR  B  ->  ( C  +  A )  <RR  ( C  +  B ) )
168, 15wi 4 1  wff  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A )  <RR  ( C  +  B
) ) )
Colors of variables: wff set class
This axiom is referenced by:  axltadd  8896
  Copyright terms: Public domain W3C validator