Theorem isumz 11700

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 2205	. . . 4 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		simp1 1000	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> M e. ZZ )
3		simp2 1001	. . . 4 |- ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) -> A C_ ( ZZ>= ` M ) )
4		c0ex 8066	. . . . . . 7 |- 0 e. _V
5	4	fvconst2 5800	. . . . . 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 2266	. . . . . . . 8 |- ( j = k -> ( j e. A <-> k e. A ) )
8	7	dcbid 840	. . . . . . 7 |- ( j = k -> (DECID j e. A <-> DECID k e. A ) )
9		simpl3 1005	. . . . . . 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 2882	. . . . . 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 3606	. . . . . 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 2241	. . . 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 1002	. . . . 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 2266	. . . . . . 7 |- ( j = a -> ( j e. A <-> a e. A ) )
17	16	dcbid 840	. . . . . 6 |- ( j = a -> (DECID j e. A <-> DECID a e. A ) )
18	17	cbvralv 2738	. . . . 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 8065	. . . 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 11695	. . 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 11605	. . . . 5 |- ~~> : dom ~~> --> CC
23		ffun 5428	. . . . 5 |- ( ~~> : dom ~~> --> CC -> Fun ~~> )
24	22, 23	ax-mp 5	. . . 4 |- Fun ~~>
25		serclim0 11616	. . . . 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 5617	. . . 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 2238	. 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 11686	. . 3 |- ( A e. Fin -> ( A = (/) \/ ( (♯ ` A ) e. NN /\ E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) ) )
31		sumeq1 11666	. . . . 5 |- ( A = (/) -> sum_ k e. A 0 = sum_ k e. (/) 0 )
32		sum0 11699	. . . . 5 |- sum_ k e. (/) 0 = 0
33	31, 32	eqtrdi 2254	. . . 4 |- ( A = (/) -> sum_ k e. A 0 = 0 )
34		eqidd 2206	. . . . . . . . 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 8065	. . . . . . . . 9 |- ( ( ( (♯ ` A ) e. NN /\ f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A ) /\ k e. A ) -> 0 e. CC )
38		elfznn 10176	. . . . . . . . . . 11 |- ( n e. ( 1 ... (♯ ` A ) ) -> n e. NN )
39	4	fvconst2 5800	. . . . . . . . . . 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 11698	. . . . . . . 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 9684	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` 1 )
44	43	fser0const 10680	. . . . . . . . . . . 12 |- ( (♯ ` A ) e. NN -> ( n e. NN |-> if ( n <_ (♯ ` A ) , ( ( NN X. { 0 } ) ` n ) , 0 ) ) = ( NN X. { 0 } ) )
45	44	seqeq3d 10600	. . . . . . . . . . 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 5578	. . . . . . . . . 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 10678	. . . . . . . . . 10 |- ( (♯ ` A ) e. NN -> ( seq 1 ( + , ( NN X. { 0 } ) ) ` (♯ ` A ) ) = 0 )
48	46, 47	eqtrd 2238	. . . . . . . . 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 2238	. . . . . . 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 1842	. . . . 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 718	. . 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 718	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   E.wex 1515   e. wcel 2176   A.wral 2484   C_ wss 3166   (/)c0 3460   ifcif 3571   {csn 3633   class class class wbr 4044   |-> cmpt 4105   X. cxp 4673   dom cdm 4675   Fun wfun 5265   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944   Fincfn 6827   CCcc 7923   0cc0 7925   1c1 7926   + caddc 7928   <_ cle 8108   NNcn 9036   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   seqcseq 10592  ♯chash 10920   ~~> cli 11589   sum_csu 11664

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-ihash 10921  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-sumdc 11665

This theorem is referenced by:  fsum00  11773  fsumdvds  12153  pcfac  12673  plymullem1  15220  nconstwlpolem0  16002