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Theorem gt0add 8147
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 948 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  0  <  ( A  +  B
) )
2 0red 7586 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  0  e.  RR )
3 simp1 946 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  A  e.  RR )
4 simp2 947 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  B  e.  RR )
53, 4readdcld 7614 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  ( A  +  B )  e.  RR )
6 axltwlin 7651 . . . 4  |-  ( ( 0  e.  RR  /\  ( A  +  B
)  e.  RR  /\  A  e.  RR )  ->  ( 0  <  ( A  +  B )  ->  ( 0  <  A  \/  A  <  ( A  +  B ) ) ) )
72, 5, 3, 6syl3anc 1181 . . 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 8103 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B
) )  ->  (
0  <  B  <->  A  <  ( A  +  B ) ) )
109orbi2d 742 . 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 667    /\ w3a 927    e. wcel 1445   class class class wbr 3867  (class class class)co 5690   RRcr 7446   0cc0 7447    + caddc 7450    < clt 7619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-i2m1 7547  ax-0id 7550  ax-rnegex 7551  ax-pre-ltwlin 7555  ax-pre-ltadd 7558
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-xp 4473  df-iota 5014  df-fv 5057  df-ov 5693  df-pnf 7621  df-mnf 7622  df-ltxr 7624
This theorem is referenced by: (None)
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