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Theorem add20 9302
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 735 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  <_  A
)
2 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  e.  RR )
3 simplll 734 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  RR )
4 addge02 9301 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  A  <->  B  <_  ( A  +  B ) ) )
52, 3, 4syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( 0  <_  A 
<->  B  <_  ( A  +  B ) ) )
61, 5mpbid 201 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  <_  ( A  +  B )
)
7 simpr 447 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  0 )
86, 7breqtrd 4063 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  <_  0
)
9 simplrr 737 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  <_  B
)
10 0re 8854 . . . . . . . . 9  |-  0  e.  RR
1110a1i 10 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  0  e.  RR )
122, 11letri3d 8977 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( B  =  0  <->  ( B  <_ 
0  /\  0  <_  B ) ) )
138, 9, 12mpbir2and 888 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  B  =  0 )
1413oveq2d 5890 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  ( A  +  0 ) )
153recnd 8877 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  CC )
1615addid1d 9028 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  + 
0 )  =  A )
1714, 7, 163eqtr3rd 2337 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  =  0 )
1817, 13jca 518 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  =  0  /\  B  =  0 ) )
1918ex 423 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  ( ( A  +  B )  =  0  ->  ( A  =  0  /\  B  =  0 ) ) )
20 oveq12 5883 . . 3  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  ( 0  +  0 ) )
21 00id 9003 . . 3  |-  ( 0  +  0 )  =  0
2220, 21syl6eq 2344 . 2  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  0 )
2319, 22impbid1 194 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    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753    + caddc 8756    <_ cle 8884
This theorem is referenced by:  add20i  9332  sumsqeq0  11198  4sqlem15  13022  4sqlem16  13023  ang180lem2  20124  mumullem2  20434  2sqlem7  20625  add20OLD  26536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889
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