Theorem fsum00 12207

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 481	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> B e. RR )
5		fsumge0.3	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> 0 <_ B )
6	5	adantlr 481	. . . . . . . . 9 |- ( ( ( ph /\ m e. A ) /\ k e. A ) -> 0 <_ B )
7		snssi 3854	. . . . . . . . . 10 |- ( m e. A -> { m } C_ A )
8	7	adantl 277	. . . . . . . . 9 |- ( ( ph /\ m e. A ) -> { m } C_ A )
9		snfig 7093	. . . . . . . . . 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 12205	. . . . . . . 8 |- ( ( ph /\ m e. A ) -> sum_ k e. { m } B <_ sum_ k e. A B )
12	11	adantlr 481	. . . . . . 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 2623	. . . . . . . . . . . 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 3180	. . . . . . . . . . . . . 14 |- F/_ k [_ m / k ]_ B
18	17	nfel1 2403	. . . . . . . . . . . . 13 |- F/ k [_ m / k ]_ B e. RR
19		nfcv 2392	. . . . . . . . . . . . . 14 |- F/_ k 0
20		nfcv 2392	. . . . . . . . . . . . . 14 |- F/_ k <_
21	19, 20, 17	nfbr 4172	. . . . . . . . . . . . 13 |- F/ k 0 <_ [_ m / k ]_ B
22	18, 21	nfan 1618	. . . . . . . . . . . 12 |- F/ k ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B )
23		csbeq1a 3156	. . . . . . . . . . . . . 14 |- ( k = m -> B = [_ m / k ]_ B )
24	23	eleq1d 2307	. . . . . . . . . . . . 13 |- ( k = m -> ( B e. RR <-> [_ m / k ]_ B e. RR ) )
25	23	breq2d 4137	. . . . . . . . . . . . 13 |- ( k = m -> ( 0 <_ B <-> 0 <_ [_ m / k ]_ B ) )
26	24, 25	anbi12d 477	. . . . . . . . . . . 12 |- ( k = m -> ( ( B e. RR /\ 0 <_ B ) <-> ( [_ m / k ]_ B e. RR /\ 0 <_ [_ m / k ]_ B ) ) )
27	22, 26	rspc 2923	. . . . . . . . . . 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 8344	. . . . . . . 8 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B e. CC )
31		sumsns 12160	. . . . . . . 8 |- ( ( m e. A /\ [_ m / k ]_ B e. CC ) -> sum_ k e. { m } B = [_ m / k ]_ B )
32	13, 30, 31	syl2anc 415	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. { m } B = [_ m / k ]_ B )
33		simplr 533	. . . . . . 7 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> sum_ k e. A B = 0 )
34	12, 32, 33	3brtr3d 4156	. . . . . 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 8316	. . . . . . 7 |- 0 e. RR
37		letri3 8396	. . . . . . 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 417	. . . . . 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 957	. . . . 5 |- ( ( ( ph /\ sum_ k e. A B = 0 ) /\ m e. A ) -> [_ m / k ]_ B = 0 )
40	39	ralrimiva 2623	. . . 4 |- ( ( ph /\ sum_ k e. A B = 0 ) -> A. m e. A [_ m / k ]_ B = 0 )
41		nfv 1581	. . . . 5 |- F/ m B = 0
42	17	nfeq1 2402	. . . . 5 |- F/ k [_ m / k ]_ B = 0
43	23	eqeq1d 2247	. . . . 5 |- ( k = m -> ( B = 0 <-> [_ m / k ]_ B = 0 ) )
44	41, 42, 43	cbvral 2782	. . . 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 12134	. . . . 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 748	. . . 4 |- ( A e. Fin -> sum_ k e. A 0 = 0 )
49		sumeq2 12103	. . . . 5 |- ( A. k e. A B = 0 -> sum_ k e. A B = sum_ k e. A 0 )
50	49	eqeq1d 2247	. . . 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 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   [_csb 3147   C_ wss 3220   {csn 3705   class class class wbr 4125   ` cfv 5372   Fincfn 7012   CCcc 8167   RRcr 8168   0cc0 8169   <_ cle 8351   ZZcz 9623   ZZ>=cuz 9900   sum_csu 12097

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-ico 10275  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by: (None)