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Theorem add20 7697
Description: Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
add20  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )

Proof of Theorem add20
StepHypRef Expression
1 simpllr 501 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  <_  A
)
2 simplrl 502 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  e.  RR )
3 simplll 500 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  RR )
4 addge02 7696 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  B  <_  ( A  +  B ) ) )
52, 3, 4syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( 0  <_  A 
<->  B  <_  ( A  +  B ) ) )
61, 5mpbid 145 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  <_  ( A  +  B )
)
7 simpr 108 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  0 )
86, 7breqtrd 3829 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  <_  0
)
9 simplrr 503 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  <_  B
)
10 0red 7234 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  e.  RR )
112, 10letri3d 7345 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( B  =  0  <->  ( B  <_ 
0  /\  0  <_  B ) ) )
128, 9, 11mpbir2and 886 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  =  0 )
1312oveq2d 5579 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  ( A  +  0 ) )
143recnd 7261 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  CC )
1514addid1d 7376 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  + 
0 )  =  A )
1613, 7, 153eqtr3rd 2124 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  =  0 )
1716, 12jca 300 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  =  0  /\  B  =  0 ) )
1817ex 113 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  +  B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
19 oveq12 5572 . . 3  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  ( 0  +  0 ) )
20 00id 7368 . . 3  |-  ( 0  +  0 )  =  0
2119, 20syl6eq 2131 . 2  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  0 )
2218, 21impbid1 140 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   RRcr 7094   0cc0 7095    + caddc 7098    <_ cle 7268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-pre-ltirr 7202  ax-pre-apti 7205  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-iota 4917  df-fv 4960  df-ov 5566  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273
This theorem is referenced by:  add20i  7712  sumsqeq0  9703
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