Theorem isumz 11279

Description: Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)

Assertion

Ref	Expression
isumz	|- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )

Distinct variable groups:   A, j, k   j, M, k

Proof of Theorem isumz

Dummy variables a f n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2157	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		simp1 982	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> M e. ZZ )
3		simp2 983	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> A C_ ( ZZ>= ` M ) )
4		c0ex 7866	. . . . . . 7 |- 0 e. _V
5	4	fvconst2 5682	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> ( ( ( ZZ>= ` M ) X. { 0 } ) ` k ) = 0 )
6	5	adantl 275	. . . . 5 |- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) /\ k e. ( ZZ>= ` M ) ) -> ( ( ( ZZ>= ` M ) X. { 0 } ) ` k ) = 0 )
7		eleq1w 2218	. . . . . . . 8 |- ( j = k -> ( j e. A <-> k e. A ) )
8	7	dcbid 824	. . . . . . 7 |- ( j = k -> (DECID j e. A <-> DECID k e. A ) )
9		simpl3 987	. . . . . . 7 |- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) /\ k e. ( ZZ>= ` M ) ) -> A. j e. ( ZZ>= ` M )DECID j e. A )
10		simpr 109	. . . . . . 7 |- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
11	8, 9, 10	rspcdva 2821	. . . . . 6 |- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )
12		ifiddc 3538	. . . . . 6 |- (DECID k e. A -> if ( k e. A , 0 , 0 ) = 0 )
13	11, 12	syl 14	. . . . 5 |- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) /\ k e. ( ZZ>= ` M ) ) -> if ( k e. A , 0 , 0 ) = 0 )
14	6, 13	eqtr4d 2193	. . . 4 |- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) /\ k e. ( ZZ>= ` M ) ) -> ( ( ( ZZ>= ` M ) X. { 0 } ) ` k ) = if ( k e. A , 0 , 0 ) )
15		simp3 984	. . . . 5 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> A. j e. ( ZZ>= ` M )DECID j e. A )
16		eleq1w 2218	. . . . . . 7 |- ( j = a -> ( j e. A <-> a e. A ) )
17	16	dcbid 824	. . . . . 6 |- ( j = a -> (DECID j e. A <-> DECID a e. A ) )
18	17	cbvralv 2680	. . . . 5 |- ( A. j e. ( ZZ>= ` M )DECID j e. A <-> A. a e. ( ZZ>= ` M )DECID a e. A )
19	15, 18	sylib 121	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> A. a e. ( ZZ>= ` M )DECID a e. A )
20		0cnd 7865	. . . 4 |- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) /\ k e. A ) -> 0 e. CC )
21	1, 2, 3, 14, 19, 20	zsumdc 11274	. . 3 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> sum_ k e. A 0 = ( ~~> ` seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ) )
22		fclim 11184	. . . . 5 |- ~~> : dom ~~> --> CC
23		ffun 5321	. . . . 5 |- ( ~~> : dom ~~> --> CC -> Fun ~~> )
24	22, 23	ax-mp 5	. . . 4 |- Fun ~~>
25		serclim0 11195	. . . . 5 |- ( M e. ZZ -> seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 )
26	2, 25	syl 14	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 )
27		funbrfv 5506	. . . 4 |- ( Fun ~~> -> ( seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 -> ( ~~> ` seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ) = 0 ) )
28	24, 26, 27	mpsyl 65	. . 3 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> ( ~~> ` seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ) = 0 )
29	21, 28	eqtrd 2190	. 2 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> sum_ k e. A 0 = 0 )
30		fz1f1o 11265	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) ) )
31		sumeq1 11245	. . . . 5 |- ( A = (/) -> sum_ k e. A 0 = sum_ k e. (/) 0 )
32		sum0 11278	. . . . 5 |- sum_ k e. (/) 0 = 0
33	31, 32	eqtrdi 2206	. . . 4 |- ( A = (/) -> sum_ k e. A 0 = 0 )
34		eqidd 2158	. . . . . . . . 9 |- ( k = ( f ` n ) -> 0 = 0 )
35		simpl 108	. . . . . . . . 9 |- ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> (♯ ` A ) e. NN )
36		simpr 109	. . . . . . . . 9 |- ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
37		0cnd 7865	. . . . . . . . 9 |- ( ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ k e. A ) -> 0 e. CC )
38		elfznn 9949	. . . . . . . . . . 11 |- ( n e. ( 1 ... (♯ ` A ) ) -> n e. NN )
39	4	fvconst2 5682	. . . . . . . . . . 11 |- ( n e. NN -> ( ( NN X. { 0 } ) ` n ) = 0 )
40	38, 39	syl 14	. . . . . . . . . 10 |- ( n e. ( 1 ... (♯ ` A ) ) -> ( ( NN X. { 0 } ) ` n ) = 0 )
41	40	adantl 275	. . . . . . . . 9 |- ( ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ n e. ( 1 ... (♯ ` A ) ) ) -> ( ( NN X. { 0 } ) ` n ) = 0 )
42	34, 35, 36, 37, 41	fsum3 11277	. . . . . . . 8 |- ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> sum_ k e. A 0 = ( seq 1 ( + , ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) ) ` (♯ ` A ) ) )
43		nnuz 9468	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` 1 )
44	43	fser0const 10408	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN -> ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) = ( NN X. { 0 } ) )
45	44	seqeq3d 10345	. . . . . . . . . . 11 |- ( (♯ ` A ) e. NN -> seq 1 ( + , ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) ) = seq 1 ( + , ( NN X. { 0 } ) ) )
46	45	fveq1d 5469	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN -> ( seq 1 ( + , ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) ) ` (♯ ` A ) ) = ( seq 1 ( + , ( NN X. { 0 } ) ) ` (♯ ` A ) ) )
47	43	ser0 10406	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN -> ( seq 1 ( + , ( NN X. { 0 } ) ) ` (♯ ` A ) ) = 0 )
48	46, 47	eqtrd 2190	. . . . . . . . 9 |- ( (♯ ` A ) e. NN -> ( seq 1 ( + , ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) ) ` (♯ ` A ) ) = 0 )
49	35, 48	syl 14	. . . . . . . 8 |- ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> ( seq 1 ( + , ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) ) ` (♯ ` A ) ) = 0 )
50	42, 49	eqtrd 2190	. . . . . . 7 |- ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> sum_ k e. A 0 = 0 )
51	50	ex 114	. . . . . 6 |- ( (♯ ` A ) e. NN -> ( f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A -> sum_ k e. A 0 = 0 ) )
52	51	exlimdv 1799	. . . . 5 |- ( (♯ ` A ) e. NN -> ( E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A -> sum_ k e. A 0 = 0 ) )
53	52	imp 123	. . . 4 |- ( ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> sum_ k e. A 0 = 0 )
54	33, 53	jaoi 706	. . 3 |- ( ( A = (/) \/ ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) ) -> sum_ k e. A 0 = 0 )
55	30, 54	syl 14	. 2 |- ( A e. Fin -> sum_ k e. A 0 = 0 )
56	29, 55	jaoi 706	1 |- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 820   /\ w3a 963   = wceq 1335   E.wex 1472   e. wcel 2128   A.wral 2435   C_ wss 3102   (/)c0 3394   ifcif 3505   {csn 3560   class class class wbr 3965   |-> cmpt 4025   X. cxp 4583   dom cdm 4585   Fun wfun 5163   -->wf 5165   -1-1-onto->wf1o 5168   ` cfv 5169  (class class class)co 5821   Fincfn 6682   CCcc 7724   0cc0 7726   1c1 7727   + caddc 7729   <_ cle 7907   NNcn 8827   ZZcz 9161   ZZ>=cuz 9433   ...cfz 9905   seqcseq 10337  ♯chash 10642   ~~> cli 11168   sum_csu 11243

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-mulrcl 7825  ax-addcom 7826  ax-mulcom 7827  ax-addass 7828  ax-mulass 7829  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-1rid 7833  ax-0id 7834  ax-rnegex 7835  ax-precex 7836  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-apti 7841  ax-pre-ltadd 7842  ax-pre-mulgt0 7843  ax-pre-mulext 7844  ax-arch 7845  ax-caucvg 7846

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-frec 6335  df-1o 6360  df-oadd 6364  df-er 6477  df-en 6683  df-dom 6684  df-fin 6685  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-reap 8444  df-ap 8451  df-div 8540  df-inn 8828  df-2 8886  df-3 8887  df-4 8888  df-n0 9085  df-z 9162  df-uz 9434  df-q 9522  df-rp 9554  df-fz 9906  df-fzo 10035  df-seqfrec 10338  df-exp 10412  df-ihash 10643  df-cj 10735  df-re 10736  df-im 10737  df-rsqrt 10891  df-abs 10892  df-clim 11169  df-sumdc 11244

This theorem is referenced by:  fsum00  11352  nconstwlpolem0  13604