Theorem fsum00 12532

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 452	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> A e. Fin )
3		fsumge0.2	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> B e. RR )
4	3	adantlr 696	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> B e. RR )
5		fsumge0.3	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> 0 <_ B )
6	5	adantlr 696	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> 0 <_ B )
7		snssi 3902	. . . . . . . . . 10 |- ( m e. A -> { m } C_ A )
8	7	adantl 453	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> { m } C_ A )
9	2, 4, 6, 8	fsumless 12530	. . . . . . . 8 |- ( ( ph /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
10	9	adantlr 696	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
11		simpr 448	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> m e. A )
12	3, 5	jca 519	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. A ) -> ( B e. RR /\ 0 <_ B ) )
13	12	ralrimiva 2749	. . . . . . . . . . . 12 |- ( ph -> A. k e. A ( B e. RR /\ 0 <_ B ) )
14	13	adantr 452	. . . . . . . . . . 11 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. k e. A ( B e. RR /\ 0 <_ B ) )
15		nfcsb1v 3243	. . . . . . . . . . . . . 14 |- F/_ k [_ m / k ]_ B
16	15	nfel1 2550	. . . . . . . . . . . . 13 |- F/ k [_ m / k ]_ B e. RR
17		nfcv 2540	. . . . . . . . . . . . . 14 |- F/_ k 0
18		nfcv 2540	. . . . . . . . . . . . . 14 |- F/_ k <_
19	17, 18, 15	nfbr 4216	. . . . . . . . . . . . 13 |- F/ k 0 <_ [_ m / k ]_ B
20	16, 19	nfan 1842	. . . . . . . . . . . 12 |- F/ k ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B )
21		csbeq1a 3219	. . . . . . . . . . . . . 14 |- ( k = m -> B = [_ m / k ]_ B )
22	21	eleq1d 2470	. . . . . . . . . . . . 13 |- ( k = m -> ( B e. RR <-> [_ m / k ]_ B e. RR ) )
23	21	breq2d 4184	. . . . . . . . . . . . 13 |- ( k = m -> ( 0 <_ B <-> 0 <_ [_ m / k ]_ B ) )
24	22, 23	anbi12d 692	. . . . . . . . . . . 12 |- ( k = m -> ( ( B e. RR /\ 0 <_ B ) <-> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) ) )
25	20, 24	rspc 3006	. . . . . . . . . . 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 456	. . . . . . . . . 10 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) )
27	26	simpld 446	. . . . . . . . 9 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. RR )
28	27	recnd 9070	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. CC )
29		sumsns 12491	. . . . . . . 8 |- ( ( m e. A /\ [_ m / k ]_ B e. CC ) -> sum_ k e. { m } B = [_ m / k ]_ B )
30	11, 28, 29	syl2anc 643	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B = [_ m / k ]_ B )
31		simplr 732	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. A B = 0 )
32	10, 30, 31	3brtr3d 4201	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B <_ 0 )
33	26	simprd 450	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> 0 <_ [_ m / k ]_ B )
34		0re 9047	. . . . . . 7 |- 0 e. RR
35		letri3 9116	. . . . . . 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 644	. . . . . 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 889	. . . . 5 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B = 0 )
38	37	ralrimiva 2749	. . . 4 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. m e. A [_ m / k ]_ B = 0 )
39		nfv 1626	. . . . 5 |- F/ m B = 0
40	15	nfeq1 2549	. . . . 5 |- F/ k [_ m / k ]_ B = 0
41	21	eqeq1d 2412	. . . . 5 |- ( k = m -> ( B = 0 <-> [_ m / k ]_ B = 0 ) )
42	39, 40, 41	cbvral 2888	. . . 4 |- ( A. k e. A B = 0 <-> A. m e. A [_ m / k ]_ B = 0 )
43	38, 42	sylibr 204	. . 3 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. k e. A B = 0 )
44	43	ex 424	. 2 |- ( ph -> ( sum_ k e. A B = 0 -> A. k e. A B = 0 ) )
45		sumz 12471	. . . . 5 |- ( ( A C_ ( ZZ>= ` 0 ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )
46	45	olcs 385	. . . 4 |- ( A e. Fin -> sum_ k e. A 0 = 0 )
47		sumeq2 12443	. . . . 5 |- ( A. k e. A B = 0 -> sum_ k e. A B = sum_ k e. A 0 )
48	47	eqeq1d 2412	. . . 4 |- ( A. k e. A B = 0 -> ( sum_ k e. A B = 0 <-> sum_ k e. A 0 = 0 ) )
49	46, 48	syl5ibrcom 214	. . 3 |- ( A e. Fin -> ( A. k e. A B = 0 -> sum_ k e. A B = 0 ) )
50	1, 49	syl 16	. 2 |- ( ph -> ( A. k e. A B = 0 -> sum_ k e. A B = 0 ) )
51	44, 50	impbid 184	1 |- ( ph -> ( sum_ k e. A B = 0 <-> A. k e. A B = 0 ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   [_csb 3211   C_ wss 3280   {csn 3774   class class class wbr 4172   ` cfv 5413   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   <_ cle 9077   ZZ>=cuz 10444   sum_csu 12434

This theorem is referenced by:  ramcl  13352  jensen  20780  eqeelen  25747  axcgrid  25759  rrnmet  26428

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435