Theorem isumz 12013

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 2231	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		simp1 1024	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> M e. ZZ )
3		simp2 1025	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> A C_ ( ZZ>= ` M ) )
4		c0ex 8216	. . . . . . 7 |- 0 e. _V
5	4	fvconst2 5878	. . . . . 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 2292	. . . . . . . 8 |- ( j = k -> ( j e. A <-> k e. A ) )
8	7	dcbid 846	. . . . . . 7 |- ( j = k -> (DECID j e. A <-> DECID k e. A ) )
9		simpl3 1029	. . . . . . 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 2916	. . . . . 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 3645	. . . . . 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 2267	. . . 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 1026	. . . . 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 2292	. . . . . . 7 |- ( j = a -> ( j e. A <-> a e. A ) )
17	16	dcbid 846	. . . . . 6 |- ( j = a -> (DECID j e. A <-> DECID a e. A ) )
18	17	cbvralv 2768	. . . . 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 8215	. . . 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 12008	. . 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 11917	. . . . 5 |- ~~> : dom ~~> --> CC
23		ffun 5492	. . . . 5 |- ( ~~> : dom ~~> --> CC -> Fun ~~> )
24	22, 23	ax-mp 5	. . . 4 |- Fun ~~>
25		serclim0 11928	. . . . 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 5691	. . . 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 2264	. 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 11998	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) ) )
31		sumeq1 11978	. . . . 5 |- ( A = (/) -> sum_ k e. A 0 = sum_ k e. (/) 0 )
32		sum0 12012	. . . . 5 |- sum_ k e. (/) 0 = 0
33	31, 32	eqtrdi 2280	. . . 4 |- ( A = (/) -> sum_ k e. A 0 = 0 )
34		eqidd 2232	. . . . . . . . 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 8215	. . . . . . . . 9 |- ( ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ k e. A ) -> 0 e. CC )
38		elfznn 10334	. . . . . . . . . . 11 |- ( n e. ( 1 ... (♯ ` A ) ) -> n e. NN )
39	4	fvconst2 5878	. . . . . . . . . . 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 12011	. . . . . . . 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 9836	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` 1 )
44	43	fser0const 10843	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN -> ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) = ( NN X. { 0 } ) )
45	44	seqeq3d 10763	. . . . . . . . . . 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 5650	. . . . . . . . . 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 10841	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN -> ( seq 1 ( + , ( NN X. { 0 } ) ) ` (♯ ` A ) ) = 0 )
48	46, 47	eqtrd 2264	. . . . . . . . 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 2264	. . . . . . 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 1867	. . . . 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 724	. . 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 724	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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2202   A.wral 2511   C_ wss 3201   (/)c0 3496   ifcif 3607   {csn 3673   class class class wbr 4093   |-> cmpt 4155   X. cxp 4729   dom cdm 4731   Fun wfun 5327   -->wf 5329   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   Fincfn 6952   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   <_ cle 8257   NNcn 9185   ZZcz 9523   ZZ>=cuz 9799   ...cfz 10288   seqcseq 10755  ♯chash 11083   ~~> cli 11901   sum_csu 11976

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-seqfrec 10756  df-exp 10847  df-ihash 11084  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-sumdc 11977

This theorem is referenced by:  fsum00  12086  fsumdvds  12466  pcfac  12986  plymullem1  15542  nconstwlpolem0  16779