Theorem fsumcvg4 29996

Description: A serie with finite support is a finite sum, and therefore converges. (Contributed by Thierry Arnoux, 6-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)

Hypotheses

Ref	Expression
fsumcvg4.s	|- S = ( ZZ>= ` M )
fsumcvg4.m	|- ( ph -> M e. ZZ )
fsumcvg4.c	|- ( ph -> F : S --> CC )
fsumcvg4.f	|- ( ph -> ( `' F " ( CC \ { 0 } ) ) e. Fin )

Assertion

Ref	Expression
fsumcvg4	|- ( ph -> seq M ( + , F ) e. dom ~~> )

Proof of Theorem fsumcvg4

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsumcvg4.s	. 2 |- S = ( ZZ>= ` M )
2		fsumcvg4.m	. 2 |- ( ph -> M e. ZZ )
3		fsumcvg4.f	. 2 |- ( ph -> ( `' F " ( CC \ { 0 } ) ) e. Fin )
4		fsumcvg4.c	. . . . 5 |- ( ph -> F : S --> CC )
5		ffun 6048	. . . . 5 |- ( F : S --> CC -> Fun F )
6		difpreima 6343	. . . . 5 |- ( Fun F -> ( `' F " ( CC \ { 0 } ) ) = ( ( `' F " CC ) \ ( `' F " { 0 } ) ) )
7	4, 5, 6	3syl 18	. . . 4 |- ( ph -> ( `' F " ( CC \ { 0 } ) ) = ( ( `' F " CC ) \ ( `' F " { 0 } ) ) )
8		difss 3737	. . . 4 |- ( ( `' F " CC ) \ ( `' F " { 0 } ) ) C_ ( `' F " CC )
9	7, 8	syl6eqss 3655	. . 3 |- ( ph -> ( `' F " ( CC \ { 0 } ) ) C_ ( `' F " CC ) )
10		fimacnv 6347	. . . 4 |- ( F : S --> CC -> ( `' F " CC ) = S )
11	4, 10	syl 17	. . 3 |- ( ph -> ( `' F " CC ) = S )
12	9, 11	sseqtrd 3641	. 2 |- ( ph -> ( `' F " ( CC \ { 0 } ) ) C_ S )
13		exmidd 432	. . . 4 |- ( ( ph /\ k e. S ) -> ( k e. ( `' F " ( CC \ { 0 } ) ) \/ -. k e. ( `' F " ( CC \ { 0 } ) ) ) )
14		eqid 2622	. . . . . . 7 |- ( F ` k ) = ( F ` k )
15	14	biantru 526	. . . . . 6 |- ( k e. ( `' F " ( CC \ { 0 } ) ) <-> ( k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = ( F ` k ) ) )
16	15	a1i 11	. . . . 5 |- ( ( ph /\ k e. S ) -> ( k e. ( `' F " ( CC \ { 0 } ) ) <-> ( k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = ( F ` k ) ) ) )
17		fvex 6201	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` M ) e. _V
18	1, 17	eqeltri 2697	. . . . . . . . . . . . . 14 |- S e. _V
19	18	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> S e. _V )
20		0nn0 11307	. . . . . . . . . . . . . 14 |- 0 e. NN0
21	20	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 0 e. NN0 )
22		eqid 2622	. . . . . . . . . . . . . 14 |- ( CC \ { 0 } ) = ( CC \ { 0 } )
23	22	ffs2 29503	. . . . . . . . . . . . 13 |- ( ( S e. _V /\ 0 e. NN0 /\ F : S --> CC ) -> ( F supp 0 ) = ( `' F " ( CC \ { 0 } ) ) )
24	19, 21, 4, 23	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( F supp 0 ) = ( `' F " ( CC \ { 0 } ) ) )
25		ffn 6045	. . . . . . . . . . . . . 14 |- ( F : S --> CC -> F Fn S )
26	4, 25	syl 17	. . . . . . . . . . . . 13 |- ( ph -> F Fn S )
27		suppvalfn 7302	. . . . . . . . . . . . 13 |- ( ( F Fn S /\ S e. _V /\ 0 e. NN0 ) -> ( F supp 0 ) = { k e. S | ( F ` k ) =/= 0 } )
28	26, 19, 21, 27	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( F supp 0 ) = { k e. S | ( F ` k ) =/= 0 } )
29	24, 28	eqtr3d 2658	. . . . . . . . . . 11 |- ( ph -> ( `' F " ( CC \ { 0 } ) ) = { k e. S | ( F ` k ) =/= 0 } )
30	29	eleq2d 2687	. . . . . . . . . 10 |- ( ph -> ( k e. ( `' F " ( CC \ { 0 } ) ) <-> k e. { k e. S | ( F ` k ) =/= 0 } ) )
31		rabid 3116	. . . . . . . . . 10 |- ( k e. { k e. S | ( F ` k ) =/= 0 } <-> ( k e. S /\ ( F ` k ) =/= 0 ) )
32	30, 31	syl6bb 276	. . . . . . . . 9 |- ( ph -> ( k e. ( `' F " ( CC \ { 0 } ) ) <-> ( k e. S /\ ( F ` k ) =/= 0 ) ) )
33	32	baibd 948	. . . . . . . 8 |- ( ( ph /\ k e. S ) -> ( k e. ( `' F " ( CC \ { 0 } ) ) <-> ( F ` k ) =/= 0 ) )
34	33	necon2bbid 2837	. . . . . . 7 |- ( ( ph /\ k e. S ) -> ( ( F ` k ) = 0 <-> -. k e. ( `' F " ( CC \ { 0 } ) ) ) )
35	34	biimprd 238	. . . . . 6 |- ( ( ph /\ k e. S ) -> ( -. k e. ( `' F " ( CC \ { 0 } ) ) -> ( F ` k ) = 0 ) )
36	35	pm4.71d 666	. . . . 5 |- ( ( ph /\ k e. S ) -> ( -. k e. ( `' F " ( CC \ { 0 } ) ) <-> ( -. k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = 0 ) ) )
37	16, 36	orbi12d 746	. . . 4 |- ( ( ph /\ k e. S ) -> ( ( k e. ( `' F " ( CC \ { 0 } ) ) \/ -. k e. ( `' F " ( CC \ { 0 } ) ) ) <-> ( ( k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = ( F ` k ) ) \/ ( -. k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = 0 ) ) ) )
38	13, 37	mpbid 222	. . 3 |- ( ( ph /\ k e. S ) -> ( ( k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = ( F ` k ) ) \/ ( -. k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = 0 ) ) )
39		eqif 4126	. . 3 |- ( ( F ` k ) = if ( k e. ( `' F " ( CC \ { 0 } ) ) , ( F ` k ) , 0 ) <-> ( ( k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = ( F ` k ) ) \/ ( -. k e. ( `' F " ( CC \ { 0 } ) ) /\ ( F ` k ) = 0 ) ) )
40	38, 39	sylibr 224	. 2 |- ( ( ph /\ k e. S ) -> ( F ` k ) = if ( k e. ( `' F " ( CC \ { 0 } ) ) , ( F ` k ) , 0 ) )
41	12	sselda 3603	. . 3 |- ( ( ph /\ k e. ( `' F " ( CC \ { 0 } ) ) ) -> k e. S )
42	4	ffvelrnda 6359	. . 3 |- ( ( ph /\ k e. S ) -> ( F ` k ) e. CC )
43	41, 42	syldan 487	. 2 |- ( ( ph /\ k e. ( `' F " ( CC \ { 0 } ) ) ) -> ( F ` k ) e. CC )
44	1, 2, 3, 12, 40, 43	fsumcvg3 14460	1 |- ( ph -> seq M ( + , F ) e. dom ~~> )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   ifcif 4086   {csn 4177   `'ccnv 5113   dom cdm 5114   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   CCcc 9934   0cc0 9936   + caddc 9939   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  eulerpartlems  30422