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Theorem fsum00 13378
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 465 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  A  e.  Fin )
3 fsumge0.2 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
43adantlr 714 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  B  e.  RR )
5 fsumge0.3 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  0  <_  B )
65adantlr 714 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  0  <_  B )
7 snssi 4124 . . . . . . . . . 10  |-  ( m  e.  A  ->  { m }  C_  A )
87adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  { m }  C_  A )
92, 4, 6, 8fsumless 13376 . . . . . . . 8  |-  ( (
ph  /\  m  e.  A )  ->  sum_ k  e.  { m } B  <_ 
sum_ k  e.  A  B )
109adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  <_  sum_ k  e.  A  B
)
11 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  m  e.  A )
123, 5jca 532 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  ( B  e.  RR  /\  0  <_  B ) )
1312ralrimiva 2829 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B )
)
1413adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B ) )
15 nfcsb1v 3410 . . . . . . . . . . . . . 14  |-  F/_ k [_ m  /  k ]_ B
1615nfel1 2631 . . . . . . . . . . . . 13  |-  F/ k
[_ m  /  k ]_ B  e.  RR
17 nfcv 2616 . . . . . . . . . . . . . 14  |-  F/_ k
0
18 nfcv 2616 . . . . . . . . . . . . . 14  |-  F/_ k  <_
1917, 18, 15nfbr 4443 . . . . . . . . . . . . 13  |-  F/ k 0  <_  [_ m  / 
k ]_ B
2016, 19nfan 1866 . . . . . . . . . . . 12  |-  F/ k ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B )
21 csbeq1a 3403 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
2221eleq1d 2523 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( B  e.  RR  <->  [_ m  / 
k ]_ B  e.  RR ) )
2321breq2d 4411 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (
0  <_  B  <->  0  <_  [_ m  /  k ]_ B ) )
2422, 23anbi12d 710 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
( B  e.  RR  /\  0  <_  B )  <->  (
[_ m  /  k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2520, 24rspc 3171 . . . . . . . . . . 11  |-  ( m  e.  A  ->  ( A. k  e.  A  ( B  e.  RR  /\  0  <_  B )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2614, 25mpan9 469 . . . . . . . . . 10  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) )
2726simpld 459 . . . . . . . . 9  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  RR )
2827recnd 9522 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
29 sumsns 13336 . . . . . . . 8  |-  ( ( m  e.  A  /\  [_ m  /  k ]_ B  e.  CC )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
3011, 28, 29syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
31 simplr 754 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  A  B  =  0 )
3210, 30, 313brtr3d 4428 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  <_  0 )
3326simprd 463 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  0  <_  [_ m  / 
k ]_ B )
34 0re 9496 . . . . . . 7  |-  0  e.  RR
35 letri3 9570 . . . . . . 7  |-  ( (
[_ m  /  k ]_ B  e.  RR  /\  0  e.  RR )  ->  ( [_ m  /  k ]_ B  =  0  <->  ( [_ m  /  k ]_ B  <_  0  /\  0  <_  [_ m  /  k ]_ B ) ) )
3627, 34, 35sylancl 662 . . . . . 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 913 . . . . 5  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  =  0
)
3837ralrimiva 2829 . . . 4  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
39 nfv 1674 . . . . 5  |-  F/ m  B  =  0
4015nfeq1 2630 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  =  0
4121eqeq1d 2456 . . . . 5  |-  ( k  =  m  ->  ( B  =  0  <->  [_ m  / 
k ]_ B  =  0 ) )
4239, 40, 41cbvral 3047 . . . 4  |-  ( A. k  e.  A  B  =  0  <->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
4338, 42sylibr 212 . . 3  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  B  = 
0 )
4443ex 434 . 2  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  ->  A. k  e.  A  B  =  0 ) )
45 sumz 13316 . . . . 5  |-  ( ( A  C_  ( ZZ>= ` 
0 )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
4645olcs 395 . . . 4  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
47 sumeq2 13288 . . . . 5  |-  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  0 )
4847eqeq1d 2456 . . . 4  |-  ( A. k  e.  A  B  =  0  ->  ( sum_ k  e.  A  B  =  0  <->  sum_ k  e.  A  0  =  0 ) )
4946, 48syl5ibrcom 222 . . 3  |-  ( A  e.  Fin  ->  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  0 ) )
501, 49syl 16 . 2  |-  ( ph  ->  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  0 ) )
5144, 50impbid 191 1  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  <->  A. k  e.  A  B  = 
0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2798   [_csb 3394    C_ wss 3435   {csn 3984   class class class wbr 4399   ` cfv 5525   Fincfn 7419   CCcc 9390   RRcr 9391   0cc0 9392    <_ cle 9529   ZZ>=cuz 10971   sum_csu 13280
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 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-ico 11416  df-fz 11554  df-fzo 11665  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281
This theorem is referenced by:  ramcl  14207  rrxcph  21027  rrxmet  21038  jensen  22514  eqeelen  23301  axcgrid  23313  rrnmet  28875
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