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Theorem gt0add 8358
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 984 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  0  <  ( A  +  B
) )
2 0red 7790 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  0  e.  RR )
3 simp1 982 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  A  e.  RR )
4 simp2 983 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  B  e.  RR )
53, 4readdcld 7818 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  ( A  +  B )  e.  RR )
6 axltwlin 7855 . . . 4  |-  ( ( 0  e.  RR  /\  ( A  +  B
)  e.  RR  /\  A  e.  RR )  ->  ( 0  <  ( A  +  B )  ->  ( 0  <  A  \/  A  <  ( A  +  B ) ) ) )
72, 5, 3, 6syl3anc 1217 . . 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 8314 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
0  <  B  <->  A  <  ( A  +  B ) ) )
109orbi2d 780 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
( 0  <  A  \/  0  <  B )  <-> 
( 0  <  A  \/  A  <  ( A  +  B ) ) ) )
118, 10mpbird 166 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 698    /\ w3a 963    e. wcel 1481   class class class wbr 3936  (class class class)co 5781   RRcr 7642   0cc0 7643    + caddc 7646    < clt 7823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-i2m1 7748  ax-0id 7751  ax-rnegex 7752  ax-pre-ltwlin 7756  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-xp 4552  df-iota 5095  df-fv 5138  df-ov 5784  df-pnf 7825  df-mnf 7826  df-ltxr 7828
This theorem is referenced by: (None)
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