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Theorem fsum00 12272
Description: A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
fsumge0.1  |-  ( ph  ->  A  e.  Fin )
fsumge0.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
fsumge0.3  |-  ( (
ph  /\  k  e.  A )  ->  0  <_  B )
Assertion
Ref Expression
fsum00  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  <->  A. k  e.  A  B  = 
0 ) )
Distinct variable groups:    A, k    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem fsum00
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 fsumge0.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  Fin )
21adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  A  e.  Fin )
3 fsumge0.2 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
43adantlr 695 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  B  e.  RR )
5 fsumge0.3 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  0  <_  B )
65adantlr 695 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  0  <_  B )
7 snssi 3775 . . . . . . . . . 10  |-  ( m  e.  A  ->  { m }  C_  A )
87adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  { m }  C_  A )
92, 4, 6, 8fsumless 12270 . . . . . . . 8  |-  ( (
ph  /\  m  e.  A )  ->  sum_ k  e.  { m } B  <_ 
sum_ k  e.  A  B )
109adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  <_  sum_ k  e.  A  B
)
11 simpr 447 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  m  e.  A )
123, 5jca 518 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  ( B  e.  RR  /\  0  <_  B ) )
1312ralrimiva 2639 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B )
)
1413adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B ) )
15 nfcsb1v 3126 . . . . . . . . . . . . . 14  |-  F/_ k [_ m  /  k ]_ B
1615nfel1 2442 . . . . . . . . . . . . 13  |-  F/ k
[_ m  /  k ]_ B  e.  RR
17 nfcv 2432 . . . . . . . . . . . . . 14  |-  F/_ k
0
18 nfcv 2432 . . . . . . . . . . . . . 14  |-  F/_ k  <_
1917, 18, 15nfbr 4083 . . . . . . . . . . . . 13  |-  F/ k 0  <_  [_ m  / 
k ]_ B
2016, 19nfan 1783 . . . . . . . . . . . 12  |-  F/ k ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B )
21 csbeq1a 3102 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
2221eleq1d 2362 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( B  e.  RR  <->  [_ m  / 
k ]_ B  e.  RR ) )
2321breq2d 4051 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (
0  <_  B  <->  0  <_  [_ m  /  k ]_ B ) )
2422, 23anbi12d 691 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
( B  e.  RR  /\  0  <_  B )  <->  (
[_ m  /  k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2520, 24rspc 2891 . . . . . . . . . . 11  |-  ( m  e.  A  ->  ( A. k  e.  A  ( B  e.  RR  /\  0  <_  B )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2614, 25mpan9 455 . . . . . . . . . 10  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) )
2726simpld 445 . . . . . . . . 9  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  RR )
2827recnd 8877 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
29 sumsns 12231 . . . . . . . 8  |-  ( ( m  e.  A  /\  [_ m  /  k ]_ B  e.  CC )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
3011, 28, 29syl2anc 642 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
31 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  A  B  =  0 )
3210, 30, 313brtr3d 4068 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  <_  0 )
3326simprd 449 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  0  <_  [_ m  / 
k ]_ B )
34 0re 8854 . . . . . . 7  |-  0  e.  RR
35 letri3 8923 . . . . . . 7  |-  ( (
[_ m  /  k ]_ B  e.  RR  /\  0  e.  RR )  ->  ( [_ m  /  k ]_ B  =  0  <->  ( [_ m  /  k ]_ B  <_  0  /\  0  <_  [_ m  /  k ]_ B ) ) )
3627, 34, 35sylancl 643 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  ( [_ m  / 
k ]_ B  =  0  <-> 
( [_ m  /  k ]_ B  <_  0  /\  0  <_  [_ m  / 
k ]_ B ) ) )
3732, 33, 36mpbir2and 888 . . . . 5  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  =  0
)
3837ralrimiva 2639 . . . 4  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
39 nfv 1609 . . . . 5  |-  F/ m  B  =  0
4015nfeq1 2441 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  =  0
4121eqeq1d 2304 . . . . 5  |-  ( k  =  m  ->  ( B  =  0  <->  [_ m  / 
k ]_ B  =  0 ) )
4239, 40, 41cbvral 2773 . . . 4  |-  ( A. k  e.  A  B  =  0  <->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
4338, 42sylibr 203 . . 3  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  B  = 
0 )
4443ex 423 . 2  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  ->  A. k  e.  A  B  =  0 ) )
45 sumz 12211 . . . . 5  |-  ( ( A  C_  ( ZZ>= ` 
0 )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
4645olcs 384 . . . 4  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
47 sumeq2 12183 . . . . 5  |-  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  0 )
4847eqeq1d 2304 . . . 4  |-  ( A. k  e.  A  B  =  0  ->  ( sum_ k  e.  A  B  =  0  <->  sum_ k  e.  A  0  =  0 ) )
4946, 48syl5ibrcom 213 . . 3  |-  ( A  e.  Fin  ->  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  0 ) )
501, 49syl 15 . 2  |-  ( ph  ->  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  0 ) )
5144, 50impbid 183 1  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  <->  A. k  e.  A  B  = 
0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   [_csb 3094    C_ wss 3165   {csn 3653   class class class wbr 4039   ` cfv 5271   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753    <_ cle 8884   ZZ>=cuz 10246   sum_csu 12174
This theorem is referenced by:  ramcl  13092  jensen  20299  eqeelen  24604  axcgrid  24616  rrnmet  26656
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  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  ax-pre-mulgt0 8830  ax-pre-sup 8831
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-reu 2563  df-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175
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