Theorem fsum00 14530

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.

Step	Hyp	Ref	Expression
1		fsumge0.1	. . . . . . . . . 10 |- ( ph -> A e. Fin )
2	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> A e. Fin )
3		fsumge0.2	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> B e. RR )
4	3	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> B e. RR )
5		fsumge0.3	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> 0 <_ B )
6	5	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> 0 <_ B )
7		snssi 4339	. . . . . . . . . 10 |- ( m e. A -> { m } C_ A )
8	7	adantl 482	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> { m } C_ A )
9	2, 4, 6, 8	fsumless 14528	. . . . . . . 8 |- ( ( ph /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
10	9	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
11		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> m e. A )
12	3, 5	jca 554	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. A ) -> ( B e. RR /\ 0 <_ B ) )
13	12	ralrimiva 2966	. . . . . . . . . . . 12 |- ( ph -> A. k e. A ( B e. RR /\ 0 <_ B ) )
14	13	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. k e. A ( B e. RR /\ 0 <_ B ) )
15		nfcsb1v 3549	. . . . . . . . . . . . . 14 |- F/_ k [_ m / k ]_ B
16	15	nfel1 2779	. . . . . . . . . . . . 13 |- F/ k [_ m / k ]_ B e. RR
17		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ k 0
18		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ k <_
19	17, 18, 15	nfbr 4699	. . . . . . . . . . . . 13 |- F/ k 0 <_ [_ m / k ]_ B
20	16, 19	nfan 1828	. . . . . . . . . . . 12 |- F/ k ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B )
21		csbeq1a 3542	. . . . . . . . . . . . . 14 |- ( k = m -> B = [_ m / k ]_ B )
22	21	eleq1d 2686	. . . . . . . . . . . . 13 |- ( k = m -> ( B e. RR <-> [_ m / k ]_ B e. RR ) )
23	21	breq2d 4665	. . . . . . . . . . . . 13 |- ( k = m -> ( 0 <_ B <-> 0 <_ [_ m / k ]_ B ) )
24	22, 23	anbi12d 747	. . . . . . . . . . . 12 |- ( k = m -> ( ( B e. RR /\ 0 <_ B ) <-> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) ) )
25	20, 24	rspc 3303	. . . . . . . . . . 11 |- ( m e. A -> ( A. k e. A ( B e. RR /\ 0 <_ B ) -> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) ) )
26	14, 25	mpan9 486	. . . . . . . . . 10 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) )
27	26	simpld 475	. . . . . . . . 9 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. RR )
28	27	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. CC )
29		sumsns 14479	. . . . . . . 8 |- ( ( m e. A /\ [_ m / k ]_ B e. CC ) -> sum_ k e. { m } B = [_ m / k ]_ B )
30	11, 28, 29	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B = [_ m / k ]_ B )
31		simplr 792	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. A B = 0 )
32	10, 30, 31	3brtr3d 4684	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B <_ 0 )
33	26	simprd 479	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> 0 <_ [_ m / k ]_ B )
34		0re 10040	. . . . . . 7 |- 0 e. RR
35		letri3 10123	. . . . . . 7 |- ( ( [_ m / k ]_ B e. RR /\ 0 e. RR ) -> ( [_ m / k ]_ B = 0 <-> ( [_ m / k ]_ B <_ 0 /\ 0 <_ [_ m / k ]_ B ) ) )
36	27, 34, 35	sylancl 694	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> ( [_ m / k ]_ B = 0 <-> ( [_ m / k ]_ B <_ 0 /\ 0 <_ [_ m / k ]_ B ) ) )
37	32, 33, 36	mpbir2and 957	. . . . 5 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B = 0 )
38	37	ralrimiva 2966	. . . 4 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. m e. A [_ m / k ]_ B = 0 )
39		nfv 1843	. . . . 5 |- F/ m B = 0
40	15	nfeq1 2778	. . . . 5 |- F/ k [_ m / k ]_ B = 0
41	21	eqeq1d 2624	. . . . 5 |- ( k = m -> ( B = 0 <-> [_ m / k ]_ B = 0 ) )
42	39, 40, 41	cbvral 3167	. . . 4 |- ( A. k e. A B = 0 <-> A. m e. A [_ m / k ]_ B = 0 )
43	38, 42	sylibr 224	. . 3 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. k e. A B = 0 )
44	43	ex 450	. 2 |- ( ph -> ( sum_ k e. A B = 0 -> A. k e. A B = 0 ) )
45		sumz 14453	. . . . 5 |- ( ( A C_ ( ZZ>= ` 0 ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )
46	45	olcs 410	. . . 4 |- ( A e. Fin -> sum_ k e. A 0 = 0 )
47		sumeq2 14424	. . . . 5 |- ( A. k e. A B = 0 -> sum_ k e. A B = sum_ k e. A 0 )
48	47	eqeq1d 2624	. . . 4 |- ( A. k e. A B = 0 -> ( sum_ k e. A B = 0 <-> sum_ k e. A 0 = 0 ) )
49	46, 48	syl5ibrcom 237	. . 3 |- ( A e. Fin -> ( A. k e. A B = 0 -> sum_ k e. A B = 0 ) )
50	1, 49	syl 17	. 2 |- ( ph -> ( A. k e. A B = 0 -> sum_ k e. A B = 0 ) )
51	44, 50	impbid 202	1 |- ( ph -> ( sum_ k e. A B = 0 <-> A. k e. A B = 0 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   [_csb 3533   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   <_ cle 10075   ZZ>=cuz 11687   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  ramcl  15733  rrxcph  23180  rrxmet  23191  jensen  24715  eqeelen  25784  axcgrid  25796  rrnmet  33628