Theorem fsum00 11973

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 variables m x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsumge0.1	. . . . . . . . . 10 |- ( ph -> A e. Fin )
2	1	adantr 276	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> A e. Fin )
3		fsumge0.2	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> B e. RR )
4	3	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> B e. RR )
5		fsumge0.3	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> 0 <_ B )
6	5	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> 0 <_ B )
7		snssi 3812	. . . . . . . . . 10 |- ( m e. A -> { m } C_ A )
8	7	adantl 277	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> { m } C_ A )
9		snfig 6967	. . . . . . . . . 10 |- ( m e. A -> { m } e. Fin )
10	9	adantl 277	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> { m } e. Fin )
11	2, 4, 6, 8, 10	fsumlessfi 11971	. . . . . . . 8 |- ( ( ph /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
12	11	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
13		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> m e. A )
14	3, 5	jca 306	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. A ) -> ( B e. RR /\ 0 <_ B ) )
15	14	ralrimiva 2603	. . . . . . . . . . . 12 |- ( ph -> A. k e. A ( B e. RR /\ 0 <_ B ) )
16	15	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. k e. A ( B e. RR /\ 0 <_ B ) )
17		nfcsb1v 3157	. . . . . . . . . . . . . 14 |- F/_ k [_ m / k ]_ B
18	17	nfel1 2383	. . . . . . . . . . . . 13 |- F/ k [_ m / k ]_ B e. RR
19		nfcv 2372	. . . . . . . . . . . . . 14 |- F/_ k 0
20		nfcv 2372	. . . . . . . . . . . . . 14 |- F/_ k <_
21	19, 20, 17	nfbr 4130	. . . . . . . . . . . . 13 |- F/ k 0 <_ [_ m / k ]_ B
22	18, 21	nfan 1611	. . . . . . . . . . . 12 |- F/ k ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B )
23		csbeq1a 3133	. . . . . . . . . . . . . 14 |- ( k = m -> B = [_ m / k ]_ B )
24	23	eleq1d 2298	. . . . . . . . . . . . 13 |- ( k = m -> ( B e. RR <-> [_ m / k ]_ B e. RR ) )
25	23	breq2d 4095	. . . . . . . . . . . . 13 |- ( k = m -> ( 0 <_ B <-> 0 <_ [_ m / k ]_ B ) )
26	24, 25	anbi12d 473	. . . . . . . . . . . 12 |- ( k = m -> ( ( B e. RR /\ 0 <_ B ) <-> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) ) )
27	22, 26	rspc 2901	. . . . . . . . . . 11 |- ( m e. A -> ( A. k e. A ( B e. RR /\ 0 <_ B ) -> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) ) )
28	16, 27	mpan9 281	. . . . . . . . . 10 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) )
29	28	simpld 112	. . . . . . . . 9 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. RR )
30	29	recnd 8175	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. CC )
31		sumsns 11926	. . . . . . . 8 |- ( ( m e. A /\ [_ m / k ]_ B e. CC ) -> sum_ k e. { m } B = [_ m / k ]_ B )
32	13, 30, 31	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B = [_ m / k ]_ B )
33		simplr 528	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. A B = 0 )
34	12, 32, 33	3brtr3d 4114	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B <_ 0 )
35	28	simprd 114	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> 0 <_ [_ m / k ]_ B )
36		0re 8146	. . . . . . 7 |- 0 e. RR
37		letri3 8227	. . . . . . 7 |- ( ( [_ m / k ]_ B e. RR /\ 0 e. RR ) -> ( [_ m / k ]_ B = 0 <-> ( [_ m / k ]_ B <_ 0 /\ 0 <_ [_ m / k ]_ B ) ) )
38	29, 36, 37	sylancl 413	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> ( [_ m / k ]_ B = 0 <-> ( [_ m / k ]_ B <_ 0 /\ 0 <_ [_ m / k ]_ B ) ) )
39	34, 35, 38	mpbir2and 950	. . . . 5 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B = 0 )
40	39	ralrimiva 2603	. . . 4 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. m e. A [_ m / k ]_ B = 0 )
41		nfv 1574	. . . . 5 |- F/ m B = 0
42	17	nfeq1 2382	. . . . 5 |- F/ k [_ m / k ]_ B = 0
43	23	eqeq1d 2238	. . . . 5 |- ( k = m -> ( B = 0 <-> [_ m / k ]_ B = 0 ) )
44	41, 42, 43	cbvral 2761	. . . 4 |- ( A. k e. A B = 0 <-> A. m e. A [_ m / k ]_ B = 0 )
45	40, 44	sylibr 134	. . 3 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. k e. A B = 0 )
46	45	ex 115	. 2 |- ( ph -> ( sum_ k e. A B = 0 -> A. k e. A B = 0 ) )
47		isumz 11900	. . . . 5 |- ( ( ( 0 e. ZZ /\ A C_ ( ZZ>= ` 0 ) /\ A. x e. ( ZZ>= ` 0 )DECID x e. A ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )
48	47	olcs 741	. . . 4 |- ( A e. Fin -> sum_ k e. A 0 = 0 )
49		sumeq2 11870	. . . . 5 |- ( A. k e. A B = 0 -> sum_ k e. A B = sum_ k e. A 0 )
50	49	eqeq1d 2238	. . . 4 |- ( A. k e. A B = 0 -> ( sum_ k e. A B = 0 <-> sum_ k e. A 0 = 0 ) )
51	48, 50	syl5ibrcom 157	. . 3 |- ( A e. Fin -> ( A. k e. A B = 0 -> sum_ k e. A B = 0 ) )
52	1, 51	syl 14	. 2 |- ( ph -> ( A. k e. A B = 0 -> sum_ k e. A B = 0 ) )
53	46, 52	impbid 129	1 |- ( ph -> ( sum_ k e. A B = 0 <-> A. k e. A B = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   [_csb 3124   C_ wss 3197   {csn 3666   class class class wbr 4083   ` cfv 5318   Fincfn 6887   CCcc 7997   RRcr 7998   0cc0 7999   <_ cle 8182   ZZcz 9446   ZZ>=cuz 9722   sum_csu 11864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-ico 10090  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865

This theorem is referenced by: (None)