Theorem clim2iser2 10176

Description: The limit of an infinite series with an initial segment added. (Contributed by Jim Kingdon, 21-Aug-2021.)

Hypotheses

Ref	Expression
clim2ser.1	|- Z = ( ZZ>= ` M )
clim2ser.2	|- ( ph -> N e. Z )
clim2ser.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
clim2ser2.5	|- ( ph -> seq ( N + 1 ) ( + , F , CC ) ~~> A )

Assertion

Ref	Expression
clim2iser2	|- ( ph -> seq M ( + , F , CC ) ~~> ( A + ( seq M ( + , F , CC ) ` N ) ) )

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

Proof of Theorem clim2iser2

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

Step	Hyp	Ref	Expression
1		eqid 2081	. 2 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2ser.2	. . . . 5 |- ( ph -> N e. Z )
3		clim2ser.1	. . . . 5 |- Z = ( ZZ>= ` M )
4	2, 3	syl6eleq 2171	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		peano2uz 8671	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
6	4, 5	syl 14	. . 3 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
7		eluzelz 8628	. . 3 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( N + 1 ) e. ZZ )
8	6, 7	syl 14	. 2 |- ( ph -> ( N + 1 ) e. ZZ )
9		clim2ser2.5	. 2 |- ( ph -> seq ( N + 1 ) ( + , F , CC ) ~~> A )
10		eluzel2 8624	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
11	4, 10	syl 14	. . . 4 |- ( ph -> M e. ZZ )
12		clim2ser.4	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
13	3, 11, 12	iserf 9453	. . 3 |- ( ph -> seq M ( + , F , CC ) : Z --> CC )
14	13, 2	ffvelrnd 5324	. 2 |- ( ph -> ( seq M ( + , F , CC ) ` N ) e. CC )
15		iseqex 9433	. . 3 |- seq M ( + , F , CC ) e. _V
16	15	a1i 9	. 2 |- ( ph -> seq M ( + , F , CC ) e. _V )
17	6, 3	syl6eleqr 2172	. . . . . 6 |- ( ph -> ( N + 1 ) e. Z )
18	3	uztrn2 8636	. . . . . 6 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
19	17, 18	sylan 277	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
20	19, 12	syldan 276	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
21	1, 8, 20	iserf 9453	. . 3 |- ( ph -> seq ( N + 1 ) ( + , F , CC ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
22	21	ffvelrnda 5323	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( + , F , CC ) ` j ) e. CC )
23		addcl 7098	. . . . 5 |- ( ( k e. CC /\ x e. CC ) -> ( k + x ) e. CC )
24	23	adantl 271	. . . 4 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k + x ) e. CC )
25		addass 7103	. . . . 5 |- ( ( k e. CC /\ x e. CC /\ y e. CC ) -> ( ( k + x ) + y ) = ( k + ( x + y ) ) )
26	25	adantl 271	. . . 4 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC /\ y e. CC ) ) -> ( ( k + x ) + y ) = ( k + ( x + y ) ) )
27		simpr 108	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. ( ZZ>= ` ( N + 1 ) ) )
28		cnex 7097	. . . . 5 |- CC e. _V
29	28	a1i 9	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> CC e. _V )
30	4	adantr 270	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
31	3	eleq2i 2145	. . . . . 6 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
32	31, 12	sylan2br 282	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
33	32	adantlr 460	. . . 4 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
34	24, 26, 27, 29, 30, 33	iseqsplit 9458	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F , CC ) ` j ) = ( ( seq M ( + , F , CC ) ` N ) + ( seq ( N + 1 ) ( + , F , CC ) ` j ) ) )
35	14	adantr 270	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F , CC ) ` N ) e. CC )
36	35, 22	addcomd 7259	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( + , F , CC ) ` N ) + ( seq ( N + 1 ) ( + , F , CC ) ` j ) ) = ( ( seq ( N + 1 ) ( + , F , CC ) ` j ) + ( seq M ( + , F , CC ) ` N ) ) )
37	34, 36	eqtrd 2113	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F , CC ) ` j ) = ( ( seq ( N + 1 ) ( + , F , CC ) ` j ) + ( seq M ( + , F , CC ) ` N ) ) )
38	1, 8, 9, 14, 16, 22, 37	climaddc1 10167	1 |- ( ph -> seq M ( + , F , CC ) ~~> ( A + ( seq M ( + , F , CC ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   CCcc 6979   1c1 6982   + caddc 6984   ZZcz 8351   ZZ>=cuz 8619   seqcseq 9431   ~~> cli 10117

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-fz 9030  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-clim 10118

This theorem is referenced by:  iiserex  10177