Theorem fsum3ser 12087

Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 12102 and fsump1 12110, which should make our notation clear and from which, along with closure fsumcl 12090, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 1-Oct-2022.)

Hypotheses

Ref	Expression
fsum3ser.1	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )
fsum3ser.2	|- ( ph -> N e. ( ZZ>= ` M ) )
fsum3ser.3	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )

Assertion

Ref	Expression
fsum3ser	|- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

Distinct variable groups:   k, F   k, M   k, N   ph, k

Allowed substitution hint:   A( k)

Proof of Theorem fsum3ser

Dummy variables m x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . . . 5 |- ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) = ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) )
2		eleq1w 2295	. . . . . 6 |- ( m = k -> ( m e. ( M ... N ) <-> k e. ( M ... N ) ) )
3		fveq2 5672	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
4	2, 3	ifbieq1d 3647	. . . . 5 |- ( m = k -> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
5		simpr 110	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
6		fsum3ser.1	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = A )
7		fsum3ser.3	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> A e. CC )
8	6, 7	eqeltrd 2311	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
9	8	adantr 276	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
10		0cnd 8269	. . . . . 6 |- ( ( ( ph /\ k e. ( ZZ>= ` M ) ) /\ -. k e. ( M ... N ) ) -> 0 e. CC )
11		eluzelz 9866	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
12		eluzel2 9861	. . . . . . 7 |- ( k e. ( ZZ>= ` M ) -> M e. ZZ )
13		fsum3ser.2	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` M ) )
14		eluzelz 9866	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
16	15	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> N e. ZZ )
17		fzdcel 10377	. . . . . . 7 |- ( ( k e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID k e. ( M ... N ) )
18	11, 12, 16, 17	syl2an23an 1336	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. ( M ... N ) )
19	9, 10, 18	ifcldadc 3654	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) e. CC )
20	1, 4, 5, 19	fvmptd3 5773	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
21	6	ifeq1d 3642	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = if ( k e. ( M ... N ) , A , 0 ) )
22	20, 21	eqtrd 2267	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , A , 0 ) )
23		elfzuz 10358	. . . 4 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
24	23, 7	sylan2 286	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
25		ssidd 3261	. . 3 |- ( ph -> ( M ... N ) C_ ( M ... N ) )
26	22, 13, 24, 18, 25	fsumsersdc 12085	. 2 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ) ` N ) )
27	23, 20	sylan2 286	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = if ( k e. ( M ... N ) , ( F ` k ) , 0 ) )
28		iftrue 3629	. . . . 5 |- ( k e. ( M ... N ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = ( F ` k ) )
29	28	adantl 277	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> if ( k e. ( M ... N ) , ( F ` k ) , 0 ) = ( F ` k ) )
30	27, 29	eqtrd 2267	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` k ) = ( F ` k ) )
31		eleq1w 2295	. . . . . 6 |- ( m = x -> ( m e. ( M ... N ) <-> x e. ( M ... N ) ) )
32		fveq2 5672	. . . . . 6 |- ( m = x -> ( F ` m ) = ( F ` x ) )
33	31, 32	ifbieq1d 3647	. . . . 5 |- ( m = x -> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) = if ( x e. ( M ... N ) , ( F ` x ) , 0 ) )
34		simpr 110	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> x e. ( ZZ>= ` M ) )
35		fveq2 5672	. . . . . . . 8 |- ( k = x -> ( F ` k ) = ( F ` x ) )
36	35	eleq1d 2303	. . . . . . 7 |- ( k = x -> ( ( F ` k ) e. CC <-> ( F ` x ) e. CC ) )
37	8	ralrimiva 2617	. . . . . . . 8 |- ( ph -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
38	37	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC )
39	36, 38, 34	rspcdva 2928	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
40		0cnd 8269	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> 0 e. CC )
41		eluzelz 9866	. . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> x e. ZZ )
42		eluzel2 9861	. . . . . . 7 |- ( x e. ( ZZ>= ` M ) -> M e. ZZ )
43	15	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> N e. ZZ )
44		fzdcel 10377	. . . . . . 7 |- ( ( x e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID x e. ( M ... N ) )
45	41, 42, 43, 44	syl2an23an 1336	. . . . . 6 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> DECID x e. ( M ... N ) )
46	39, 40, 45	ifcldcd 3662	. . . . 5 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> if ( x e. ( M ... N ) , ( F ` x ) , 0 ) e. CC )
47	1, 33, 34, 46	fvmptd3 5773	. . . 4 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` x ) = if ( x e. ( M ... N ) , ( F ` x ) , 0 ) )
48	47, 46	eqeltrd 2311	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ` x ) e. CC )
49	36	cbvralv 2780	. . . . 5 |- ( A. k e. ( ZZ>= ` M ) ( F ` k ) e. CC <-> A. x e. ( ZZ>= ` M ) ( F ` x ) e. CC )
50	37, 49	sylib 122	. . . 4 |- ( ph -> A. x e. ( ZZ>= ` M ) ( F ` x ) e. CC )
51	50	r19.21bi 2632	. . 3 |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. CC )
52		addcl 8254	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
53	52	adantl 277	. . 3 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
54	13, 30, 48, 51, 53	seq3fveq 10845	. 2 |- ( ph -> ( seq M ( + , ( m e. ( ZZ>= ` M ) |-> if ( m e. ( M ... N ) , ( F ` m ) , 0 ) ) ) ` N ) = ( seq M ( + , F ) ` N ) )
55	26, 54	eqtrd 2267	1 |- ( ph -> sum_ k e. ( M ... N ) A = ( seq M ( + , F ) ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   ifcif 3622   |-> cmpt 4173   ` cfv 5354  (class class class)co 6052   CCcc 8127   0cc0 8129   + caddc 8132   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345   seqcseq 10813   sum_csu 12042

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-sumdc 12043

This theorem is referenced by:  isumclim3  12113  iserabs  12165  isumsplit  12181  trireciplem  12190  geolim  12201  geo2lim  12206  cvgratnnlemseq  12216  mertenslem2  12226  mertensabs  12227  efcvgfsum  12357  effsumlt  12382  cvgcmp2nlemabs  16833