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Theorem add20 8431
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 534 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  <_  A
)
2 simplrl 535 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  e.  RR )
3 simplll 533 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  RR )
4 addge02 8430 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  B  <_  ( A  +  B ) ) )
52, 3, 4syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( 0  <_  A 
<->  B  <_  ( A  +  B ) ) )
61, 5mpbid 147 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  <_  ( A  +  B )
)
7 simpr 110 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  0 )
86, 7breqtrd 4030 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  <_  0
)
9 simplrr 536 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  <_  B
)
10 0red 7958 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  e.  RR )
112, 10letri3d 8073 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( B  =  0  <->  ( B  <_ 
0  /\  0  <_  B ) ) )
128, 9, 11mpbir2and 944 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  =  0 )
1312oveq2d 5891 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  ( A  +  0 ) )
143recnd 7986 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  CC )
1514addid1d 8106 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  + 
0 )  =  A )
1613, 7, 153eqtr3rd 2219 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  =  0 )
1716, 12jca 306 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  =  0  /\  B  =  0 ) )
1817ex 115 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  +  B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
19 oveq12 5884 . . 3  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  ( 0  +  0 ) )
20 00id 8098 . . 3  |-  ( 0  +  0 )  =  0
2119, 20eqtrdi 2226 . 2  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  0 )
2218, 21impbid1 142 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4004  (class class class)co 5875   RRcr 7810   0cc0 7811    + caddc 7814    <_ cle 7993
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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-apti 7926  ax-pre-ltadd 7927
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 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-cnv 4635  df-iota 5179  df-fv 5225  df-ov 5878  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998
This theorem is referenced by:  add20i  8449  xnn0xadd0  9867  sumsqeq0  10599  2sqlem7  14471
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