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Theorem addgt0 8395
Description: The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
addgt0  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  +  B
) )

Proof of Theorem addgt0
StepHypRef Expression
1 00id 8088 . 2  |-  ( 0  +  0 )  =  0
2 0re 7948 . . . 4  |-  0  e.  RR
3 lt2add 8392 . . . 4  |-  ( ( ( 0  e.  RR  /\  0  e.  RR )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( ( 0  < 
A  /\  0  <  B )  ->  ( 0  +  0 )  < 
( A  +  B
) ) )
42, 2, 3mpanl12 436 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  ( 0  +  0 )  < 
( A  +  B
) ) )
54imp 124 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  (
0  +  0 )  <  ( A  +  B ) )
61, 5eqbrtrrid 4036 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  +  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   class class class wbr 4000  (class class class)co 5869   RRcr 7801   0cc0 7802    + caddc 7805    < clt 7982
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4629  df-iota 5174  df-fv 5220  df-ov 5872  df-pnf 7984  df-mnf 7985  df-ltxr 7987
This theorem is referenced by:  addgt0i  8435  rpaddcl  9664
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