Theorem isumz 11915

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 2229	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		simp1 1021	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> M e. ZZ )
3		simp2 1022	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> A C_ ( ZZ>= ` M ) )
4		c0ex 8151	. . . . . . 7 |- 0 e. _V
5	4	fvconst2 5859	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> ( ( ( ZZ>= ` M ) X. { 0 } ) ` k ) = 0 )
6	5	adantl 277	. . . . 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 2290	. . . . . . . 8 |- ( j = k -> ( j e. A <-> k e. A ) )
8	7	dcbid 843	. . . . . . 7 |- ( j = k -> (DECID j e. A <-> DECID k e. A ) )
9		simpl3 1026	. . . . . . 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 110	. . . . . . 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 2912	. . . . . 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 3638	. . . . . 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 2265	. . . 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 1023	. . . . 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 2290	. . . . . . 7 |- ( j = a -> ( j e. A <-> a e. A ) )
17	16	dcbid 843	. . . . . 6 |- ( j = a -> (DECID j e. A <-> DECID a e. A ) )
18	17	cbvralv 2765	. . . . 5 |- ( A. j e. ( ZZ>= ` M )DECID j e. A <-> A. a e. ( ZZ>= ` M )DECID a e. A )
19	15, 18	sylib 122	. . . 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 8150	. . . 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 11910	. . 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 11820	. . . . 5 |- ~~> : dom ~~> --> CC
23		ffun 5476	. . . . 5 |- ( ~~> : dom ~~> --> CC -> Fun ~~> )
24	22, 23	ax-mp 5	. . . 4 |- Fun ~~>
25		serclim0 11831	. . . . 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 5672	. . . 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 2262	. 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 11901	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) ) )
31		sumeq1 11881	. . . . 5 |- ( A = (/) -> sum_ k e. A 0 = sum_ k e. (/) 0 )
32		sum0 11914	. . . . 5 |- sum_ k e. (/) 0 = 0
33	31, 32	eqtrdi 2278	. . . 4 |- ( A = (/) -> sum_ k e. A 0 = 0 )
34		eqidd 2230	. . . . . . . . 9 |- ( k = ( f ` n ) -> 0 = 0 )
35		simpl 109	. . . . . . . . 9 |- ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> (♯ ` A ) e. NN )
36		simpr 110	. . . . . . . . 9 |- ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
37		0cnd 8150	. . . . . . . . 9 |- ( ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ k e. A ) -> 0 e. CC )
38		elfznn 10262	. . . . . . . . . . 11 |- ( n e. ( 1 ... (♯ ` A ) ) -> n e. NN )
39	4	fvconst2 5859	. . . . . . . . . . 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 277	. . . . . . . . 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 11913	. . . . . . . 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 9770	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` 1 )
44	43	fser0const 10769	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN -> ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) = ( NN X. { 0 } ) )
45	44	seqeq3d 10689	. . . . . . . . . . 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 5631	. . . . . . . . . 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 10767	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN -> ( seq 1 ( + , ( NN X. { 0 } ) ) ` (♯ ` A ) ) = 0 )
48	46, 47	eqtrd 2262	. . . . . . . . 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 2262	. . . . . . 7 |- ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> sum_ k e. A 0 = 0 )
51	50	ex 115	. . . . . 6 |- ( (♯ ` A ) e. NN -> ( f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A -> sum_ k e. A 0 = 0 ) )
52	51	exlimdv 1865	. . . . 5 |- ( (♯ ` A ) e. NN -> ( E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A -> sum_ k e. A 0 = 0 ) )
53	52	imp 124	. . . 4 |- ( ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) -> sum_ k e. A 0 = 0 )
54	33, 53	jaoi 721	. . 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 721	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 104   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   C_ wss 3197   (/)c0 3491   ifcif 3602   {csn 3666   class class class wbr 4083   |-> cmpt 4145   X. cxp 4717   dom cdm 4719   Fun wfun 5312   -->wf 5314   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6007   Fincfn 6895   CCcc 8008   0cc0 8010   1c1 8011   + caddc 8013   <_ cle 8193   NNcn 9121   ZZcz 9457   ZZ>=cuz 9733   ...cfz 10216   seqcseq 10681  ♯chash 11009   ~~> cli 11804   sum_csu 11879

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-seqfrec 10682  df-exp 10773  df-ihash 11010  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-clim 11805  df-sumdc 11880

This theorem is referenced by:  fsum00  11988  fsumdvds  12368  pcfac  12888  plymullem1  15437  nconstwlpolem0  16491