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Theorem add20 9954
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 758 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  <_  A
)
2 simplrl 759 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  e.  RR )
3 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  RR )
4 addge02 9953 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  B  <_  ( A  +  B ) ) )
52, 3, 4syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( 0  <_  A 
<->  B  <_  ( A  +  B ) ) )
61, 5mpbid 210 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  <_  ( A  +  B )
)
7 simpr 461 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  0 )
86, 7breqtrd 4416 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  <_  0
)
9 simplrr 760 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  <_  B
)
10 0red 9490 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  e.  RR )
112, 10letri3d 9619 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( B  =  0  <->  ( B  <_ 
0  /\  0  <_  B ) ) )
128, 9, 11mpbir2and 913 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  =  0 )
1312oveq2d 6208 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  ( A  +  0 ) )
143recnd 9515 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  CC )
1514addid1d 9672 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  + 
0 )  =  A )
1613, 7, 153eqtr3rd 2501 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  =  0 )
1716, 12jca 532 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  =  0  /\  B  =  0 ) )
1817ex 434 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  +  B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
19 oveq12 6201 . . 3  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  ( 0  +  0 ) )
20 00id 9647 . . 3  |-  ( 0  +  0 )  =  0
2119, 20syl6eq 2508 . 2  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  0 )
2218, 21impbid1 203 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  +  B )  =  0  <->  ( A  =  0  /\  B  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4392  (class class class)co 6192   RRcr 9384   0cc0 9385    + caddc 9388    <_ cle 9522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527
This theorem is referenced by:  add20i  9986  sumsqeq0  12047  4sqlem15  14124  4sqlem16  14125  ang180lem2  22324  mumullem2  22636  2sqlem7  22827
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