ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  gt0add Unicode version

Theorem gt0add 7638
Description: A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)
Assertion
Ref Expression
gt0add  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
0  <  A  \/  0  <  B ) )

Proof of Theorem gt0add
StepHypRef Expression
1 simp3 917 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  0  <  ( A  +  B
) )
2 0red 7086 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  0  e.  RR )
3 simp1 915 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  A  e.  RR )
4 simp2 916 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  B  e.  RR )
53, 4readdcld 7114 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  ( A  +  B )  e.  RR )
6 axltwlin 7146 . . . 4  |-  ( ( 0  e.  RR  /\  ( A  +  B
)  e.  RR  /\  A  e.  RR )  ->  ( 0  <  ( A  +  B )  ->  ( 0  <  A  \/  A  <  ( A  +  B ) ) ) )
72, 5, 3, 6syl3anc 1146 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
0  <  ( A  +  B )  ->  (
0  <  A  \/  A  <  ( A  +  B ) ) ) )
81, 7mpd 13 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
0  <  A  \/  A  <  ( A  +  B ) ) )
94, 3ltaddposd 7594 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
0  <  B  <->  A  <  ( A  +  B ) ) )
109orbi2d 714 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
( 0  <  A  \/  0  <  B )  <-> 
( 0  <  A  \/  A  <  ( A  +  B ) ) ) )
118, 10mpbird 160 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
0  <  A  \/  0  <  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 639    /\ w3a 896    e. wcel 1409   class class class wbr 3792  (class class class)co 5540   RRcr 6946   0cc0 6947    + caddc 6950    < clt 7119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-pre-ltwlin 7055  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-xp 4379  df-iota 4895  df-fv 4938  df-ov 5543  df-pnf 7121  df-mnf 7122  df-ltxr 7124
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator