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Theorem add20 9286
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 9285 . . . . . . . . . 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 4047 . . . . . . 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 8838 . . . . . . . . 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 8961 . . . . . . 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 5874 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  +  B )  =  ( A  +  0 ) )
153recnd 8861 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  A  e.  CC )
1615addid1d 9012 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  /\  ( A  +  B
)  =  0 )  ->  ( A  + 
0 )  =  A )
1714, 7, 163eqtr3rd 2324 . . . 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 5867 . . 3  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  B )  =  ( 0  +  0 ) )
21 00id 8987 . . 3  |-  ( 0  +  0 )  =  0
2220, 21syl6eq 2331 . 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 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737    + caddc 8740    <_ cle 8868
This theorem is referenced by:  add20i  9316  sumsqeq0  11182  4sqlem15  13006  4sqlem16  13007  ang180lem2  20108  mumullem2  20418  2sqlem7  20609  add20OLD  26433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873
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