Theorem fsum00 11263

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 274	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> A e. Fin )
3		fsumge0.2	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> B e. RR )
4	3	adantlr 469	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> B e. RR )
5		fsumge0.3	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> 0 <_ B )
6	5	adantlr 469	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> 0 <_ B )
7		snssi 3672	. . . . . . . . . 10 |- ( m e. A -> { m } C_ A )
8	7	adantl 275	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> { m } C_ A )
9		snfig 6716	. . . . . . . . . 10 |- ( m e. A -> { m } e. Fin )
10	9	adantl 275	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> { m } e. Fin )
11	2, 4, 6, 8, 10	fsumlessfi 11261	. . . . . . . 8 |- ( ( ph /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
12	11	adantlr 469	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
13		simpr 109	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> m e. A )
14	3, 5	jca 304	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. A ) -> ( B e. RR /\ 0 <_ B ) )
15	14	ralrimiva 2508	. . . . . . . . . . . 12 |- ( ph -> A. k e. A ( B e. RR /\ 0 <_ B ) )
16	15	adantr 274	. . . . . . . . . . 11 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. k e. A ( B e. RR /\ 0 <_ B ) )
17		nfcsb1v 3040	. . . . . . . . . . . . . 14 |- F/_ k [_ m / k ]_ B
18	17	nfel1 2293	. . . . . . . . . . . . 13 |- F/ k [_ m / k ]_ B e. RR
19		nfcv 2282	. . . . . . . . . . . . . 14 |- F/_ k 0
20		nfcv 2282	. . . . . . . . . . . . . 14 |- F/_ k <_
21	19, 20, 17	nfbr 3982	. . . . . . . . . . . . 13 |- F/ k 0 <_ [_ m / k ]_ B
22	18, 21	nfan 1545	. . . . . . . . . . . 12 |- F/ k ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B )
23		csbeq1a 3016	. . . . . . . . . . . . . 14 |- ( k = m -> B = [_ m / k ]_ B )
24	23	eleq1d 2209	. . . . . . . . . . . . 13 |- ( k = m -> ( B e. RR <-> [_ m / k ]_ B e. RR ) )
25	23	breq2d 3949	. . . . . . . . . . . . 13 |- ( k = m -> ( 0 <_ B <-> 0 <_ [_ m / k ]_ B ) )
26	24, 25	anbi12d 465	. . . . . . . . . . . 12 |- ( k = m -> ( ( B e. RR /\ 0 <_ B ) <-> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) ) )
27	22, 26	rspc 2787	. . . . . . . . . . 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 279	. . . . . . . . . 10 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) )
29	28	simpld 111	. . . . . . . . 9 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. RR )
30	29	recnd 7818	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. CC )
31		sumsns 11216	. . . . . . . 8 |- ( ( m e. A /\ [_ m / k ]_ B e. CC ) -> sum_ k e. { m } B = [_ m / k ]_ B )
32	13, 30, 31	syl2anc 409	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B = [_ m / k ]_ B )
33		simplr 520	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. A B = 0 )
34	12, 32, 33	3brtr3d 3967	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B <_ 0 )
35	28	simprd 113	. . . . . 6 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> 0 <_ [_ m / k ]_ B )
36		0re 7790	. . . . . . 7 |- 0 e. RR
37		letri3 7869	. . . . . . 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 410	. . . . . 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 929	. . . . 5 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B = 0 )
40	39	ralrimiva 2508	. . . 4 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. m e. A [_ m / k ]_ B = 0 )
41		nfv 1509	. . . . 5 |- F/ m B = 0
42	17	nfeq1 2292	. . . . 5 |- F/ k [_ m / k ]_ B = 0
43	23	eqeq1d 2149	. . . . 5 |- ( k = m -> ( B = 0 <-> [_ m / k ]_ B = 0 ) )
44	41, 42, 43	cbvral 2653	. . . 4 |- ( A. k e. A B = 0 <-> A. m e. A [_ m / k ]_ B = 0 )
45	40, 44	sylibr 133	. . 3 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. k e. A B = 0 )
46	45	ex 114	. 2 |- ( ph -> ( sum_ k e. A B = 0 -> A. k e. A B = 0 ) )
47		isumz 11190	. . . . 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 726	. . . 4 |- ( A e. Fin -> sum_ k e. A 0 = 0 )
49		sumeq2 11160	. . . . 5 |- ( A. k e. A B = 0 -> sum_ k e. A B = sum_ k e. A 0 )
50	49	eqeq1d 2149	. . . 4 |- ( A. k e. A B = 0 -> ( sum_ k e. A B = 0 <-> sum_ k e. A 0 = 0 ) )
51	48, 50	syl5ibrcom 156	. . 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 128	1 |- ( ph -> ( sum_ k e. A B = 0 <-> A. k e. A B = 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 820   /\ w3a 963   = wceq 1332   e. wcel 1481   A.wral 2417   [_csb 3007   C_ wss 3076   {csn 3532   class class class wbr 3937   ` cfv 5131   Fincfn 6642   CCcc 7642   RRcr 7643   0cc0 7644   <_ cle 7825   ZZcz 9078   ZZ>=cuz 9350   sum_csu 11154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-ico 9707  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155

This theorem is referenced by: (None)