Theorem isumz 12134

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 2238	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		simp1 1028	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> M e. ZZ )
3		simp2 1029	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> A C_ ( ZZ>= ` M ) )
4		c0ex 8310	. . . . . . 7 |- 0 e. _V
5	4	fvconst2 5922	. . . . . 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 2299	. . . . . . . 8 |- ( j = k -> ( j e. A <-> k e. A ) )
8	7	dcbid 850	. . . . . . 7 |- ( j = k -> (DECID j e. A <-> DECID k e. A ) )
9		simpl3 1033	. . . . . . 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 2934	. . . . . 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 3673	. . . . . 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 2274	. . . 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 1030	. . . . 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 2299	. . . . . . 7 |- ( j = a -> ( j e. A <-> a e. A ) )
17	16	dcbid 850	. . . . . 6 |- ( j = a -> (DECID j e. A <-> DECID a e. A ) )
18	17	cbvralv 2786	. . . . 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 8309	. . . 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 12129	. . 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 12038	. . . . 5 |- ~~> : dom ~~> --> CC
23		ffun 5531	. . . . 5 |- ( ~~> : dom ~~> --> CC -> Fun ~~> )
24	22, 23	ax-mp 5	. . . 4 |- Fun ~~>
25		serclim0 12049	. . . . 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 5733	. . . 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 2271	. 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 12119	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) ) )
31		sumeq1 12099	. . . . 5 |- ( A = (/) -> sum_ k e. A 0 = sum_ k e. (/) 0 )
32		sum0 12133	. . . . 5 |- sum_ k e. (/) 0 = 0
33	31, 32	eqtrdi 2287	. . . 4 |- ( A = (/) -> sum_ k e. A 0 = 0 )
34		eqidd 2239	. . . . . . . . 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 8309	. . . . . . . . 9 |- ( ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ k e. A ) -> 0 e. CC )
38		elfznn 10438	. . . . . . . . . . 11 |- ( n e. ( 1 ... (♯ ` A ) ) -> n e. NN )
39	4	fvconst2 5922	. . . . . . . . . . 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 12132	. . . . . . . 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 9937	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` 1 )
44	43	fser0const 10950	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN -> ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) = ( NN X. { 0 } ) )
45	44	seqeq3d 10870	. . . . . . . . . . 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 5692	. . . . . . . . . 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 10948	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN -> ( seq 1 ( + , ( NN X. { 0 } ) ) ` (♯ ` A ) ) = 0 )
48	46, 47	eqtrd 2271	. . . . . . . . 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 2271	. . . . . . 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 1872	. . . . 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 728	. . 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 728	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 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   C_ wss 3220   (/)c0 3520   ifcif 3635   {csn 3705   class class class wbr 4125   |-> cmpt 4187   X. cxp 4767   dom cdm 4769   Fun wfun 5366   -->wf 5368   -1-1-onto->wf1o 5371   ` cfv 5372  (class class class)co 6075   Fincfn 7012   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   <_ cle 8351   NNcn 9283   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862  ♯chash 11192   ~~> cli 12022   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-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:  fsum00  12207  fsumdvds  12587  pcfac  13107  plymullem1  15772  nconstwlpolem0  17018