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Theorem gt0add 8561
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 1001 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  0  <  ( A  +  B
) )
2 0red 7989 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  0  e.  RR )
3 simp1 999 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  A  e.  RR )
4 simp2 1000 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  B  e.  RR )
53, 4readdcld 8018 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  ( A  +  B )  e.  RR )
6 axltwlin 8056 . . . 4  |-  ( ( 0  e.  RR  /\  ( A  +  B
)  e.  RR  /\  A  e.  RR )  ->  ( 0  <  ( A  +  B )  ->  ( 0  <  A  \/  A  <  ( A  +  B ) ) ) )
72, 5, 3, 6syl3anc 1249 . . 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 8517 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
0  <  B  <->  A  <  ( A  +  B ) ) )
109orbi2d 791 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
( 0  <  A  \/  0  <  B )  <-> 
( 0  <  A  \/  A  <  ( A  +  B ) ) ) )
118, 10mpbird 167 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 709    /\ w3a 980    e. wcel 2160   class class class wbr 4018  (class class class)co 5897   RRcr 7841   0cc0 7842    + caddc 7845    < clt 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-i2m1 7947  ax-0id 7950  ax-rnegex 7951  ax-pre-ltwlin 7955  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-iota 5196  df-fv 5243  df-ov 5900  df-pnf 8025  df-mnf 8026  df-ltxr 8028
This theorem is referenced by: (None)
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