Theorem clim2ser 11300

Description: The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)

Hypotheses

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

Assertion

Ref	Expression
clim2ser	|- ( ph -> seq ( N + 1 ) ( + , F ) ~~> ( A - ( seq M ( + , F ) ` N ) ) )

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

Proof of Theorem clim2ser

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

Step	Hyp	Ref	Expression
1		eqid 2170	. 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	eleqtrdi 2263	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		peano2uz 9542	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
6	4, 5	syl 14	. . 3 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
7		eluzelz 9496	. . 3 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( N + 1 ) e. ZZ )
8	6, 7	syl 14	. 2 |- ( ph -> ( N + 1 ) e. ZZ )
9		clim2ser.5	. 2 |- ( ph -> seq M ( + , F ) ~~> A )
10		eluzel2 9492	. . . . 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	serf 10430	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
14	13, 2	ffvelrnd 5632	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) e. CC )
15		seqex 10403	. . 3 |- seq ( N + 1 ) ( + , F ) e. _V
16	15	a1i 9	. 2 |- ( ph -> seq ( N + 1 ) ( + , F ) e. _V )
17	13	adantr 274	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> seq M ( + , F ) : Z --> CC )
18	6, 3	eleqtrrdi 2264	. . . 4 |- ( ph -> ( N + 1 ) e. Z )
19	3	uztrn2 9504	. . . 4 |- ( ( ( N + 1 ) e. Z /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
20	18, 19	sylan 281	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
21	17, 20	ffvelrnd 5632	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) e. CC )
22		addcl 7899	. . . . . 6 |- ( ( k e. CC /\ x e. CC ) -> ( k + x ) e. CC )
23	22	adantl 275	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k + x ) e. CC )
24		addass 7904	. . . . . 6 |- ( ( k e. CC /\ x e. CC /\ y e. CC ) -> ( ( k + x ) + y ) = ( k + ( x + y ) ) )
25	24	adantl 275	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC /\ y e. CC ) ) -> ( ( k + x ) + y ) = ( k + ( x + y ) ) )
26		simpr 109	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. ( ZZ>= ` ( N + 1 ) ) )
27	4	adantr 274	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
28	3	eleq2i 2237	. . . . . . 7 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
29	28, 12	sylan2br 286	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
30	29	adantlr 474	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
31	23, 25, 26, 27, 30	seq3split 10435	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) = ( ( seq M ( + , F ) ` N ) + ( seq ( N + 1 ) ( + , F ) ` j ) ) )
32	31	oveq1d 5868	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( seq M ( + , F ) ` j ) - ( seq M ( + , F ) ` N ) ) = ( ( ( seq M ( + , F ) ` N ) + ( seq ( N + 1 ) ( + , F ) ` j ) ) - ( seq M ( + , F ) ` N ) ) )
33	14	adantr 274	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` N ) e. CC )
34	3	uztrn2 9504	. . . . . . . 8 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
35	18, 34	sylan 281	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
36	35, 12	syldan 280	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
37	1, 8, 36	serf 10430	. . . . 5 |- ( ph -> seq ( N + 1 ) ( + , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
38	37	ffvelrnda 5631	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( + , F ) ` j ) e. CC )
39	33, 38	pncan2d 8232	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( ( ( seq M ( + , F ) ` N ) + ( seq ( N + 1 ) ( + , F ) ` j ) ) - ( seq M ( + , F ) ` N ) ) = ( seq ( N + 1 ) ( + , F ) ` j ) )
40	32, 39	eqtr2d 2204	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( + , F ) ` j ) = ( ( seq M ( + , F ) ` j ) - ( seq M ( + , F ) ` N ) ) )
41	1, 8, 9, 14, 16, 21, 40	climsubc1 11295	1 |- ( ph -> seq ( N + 1 ) ( + , F ) ~~> ( A - ( seq M ( + , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   CCcc 7772   1c1 7775   + caddc 7777   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   seqcseq 10401   ~~> cli 11241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-fz 9966  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242

This theorem is referenced by:  iserex  11302  ege2le3  11634