Theorem clim2ser 11278

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 2165	. 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 2259	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		peano2uz 9521	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
6	4, 5	syl 14	. . 3 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
7		eluzelz 9475	. . 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 9471	. . . . 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 10409	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
14	13, 2	ffvelrnd 5621	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) e. CC )
15		seqex 10382	. . 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 2260	. . . 4 |- ( ph -> ( N + 1 ) e. Z )
19	3	uztrn2 9483	. . . 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 5621	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) e. CC )
22		addcl 7878	. . . . . 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 7883	. . . . . 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 2233	. . . . . . 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 469	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
31	23, 25, 26, 27, 30	seq3split 10414	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) = ( ( seq M ( + , F ) ` N ) + ( seq ( N + 1 ) ( + , F ) ` j ) ) )
32	31	oveq1d 5857	. . 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 9483	. . . . . . . 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 10409	. . . . 5 |- ( ph -> seq ( N + 1 ) ( + , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
38	37	ffvelrnda 5620	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( + , F ) ` j ) e. CC )
39	33, 38	pncan2d 8211	. . 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 2199	. 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 11273	1 |- ( ph -> seq ( N + 1 ) ( + , F ) ~~> ( A - ( seq M ( + , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   CCcc 7751   1c1 7754   + caddc 7756   - cmin 8069   ZZcz 9191   ZZ>=cuz 9466   seqcseq 10380   ~~> cli 11219

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-fz 9945  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220

This theorem is referenced by:  iserex  11280  ege2le3  11612