Theorem clim2ser 11918

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 2231	. 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 2324	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		peano2uz 9820	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
6	4, 5	syl 14	. . 3 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
7		eluzelz 9768	. . 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 9763	. . . . 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 10749	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
14	13, 2	ffvelcdmd 5784	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) e. CC )
15		seqex 10715	. . 3 |- seq ( N + 1 ) ( + , F ) e. _V
16	15	a1i 9	. 2 |- ( ph -> seq ( N + 1 ) ( + , F ) e. _V )
17	13	adantr 276	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> seq M ( + , F ) : Z --> CC )
18	6, 3	eleqtrrdi 2325	. . . 4 |- ( ph -> ( N + 1 ) e. Z )
19	3	uztrn2 9777	. . . 4 |- ( ( ( N + 1 ) e. Z /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
20	18, 19	sylan 283	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. Z )
21	17, 20	ffvelcdmd 5784	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) e. CC )
22		addcl 8160	. . . . . 6 |- ( ( k e. CC /\ x e. CC ) -> ( k + x ) e. CC )
23	22	adantl 277	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k + x ) e. CC )
24		addass 8165	. . . . . 6 |- ( ( k e. CC /\ x e. CC /\ y e. CC ) -> ( ( k + x ) + y ) = ( k + ( x + y ) ) )
25	24	adantl 277	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC /\ y e. CC ) ) -> ( ( k + x ) + y ) = ( k + ( x + y ) ) )
26		simpr 110	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. ( ZZ>= ` ( N + 1 ) ) )
27	4	adantr 276	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
28	3	eleq2i 2298	. . . . . . 7 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
29	28, 12	sylan2br 288	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
30	29	adantlr 477	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
31	23, 25, 26, 27, 30	seq3split 10754	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) = ( ( seq M ( + , F ) ` N ) + ( seq ( N + 1 ) ( + , F ) ` j ) ) )
32	31	oveq1d 6036	. . 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 276	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` N ) e. CC )
34	3	uztrn2 9777	. . . . . . . 8 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
35	18, 34	sylan 283	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
36	35, 12	syldan 282	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
37	1, 8, 36	serf 10749	. . . . 5 |- ( ph -> seq ( N + 1 ) ( + , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
38	37	ffvelcdmda 5783	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( + , F ) ` j ) e. CC )
39	33, 38	pncan2d 8495	. . 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 2265	. 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 11913	1 |- ( ph -> seq ( N + 1 ) ( + , F ) ~~> ( A - ( seq M ( + , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   -->wf 5322   ` cfv 5326  (class class class)co 6021   CCcc 8033   1c1 8036   + caddc 8038   - cmin 8353   ZZcz 9482   ZZ>=cuz 9758   seqcseq 10713   ~~> cli 11859

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8126  ax-resscn 8127  ax-1cn 8128  ax-1re 8129  ax-icn 8130  ax-addcl 8131  ax-addrcl 8132  ax-mulcl 8133  ax-mulrcl 8134  ax-addcom 8135  ax-mulcom 8136  ax-addass 8137  ax-mulass 8138  ax-distr 8139  ax-i2m1 8140  ax-0lt1 8141  ax-1rid 8142  ax-0id 8143  ax-rnegex 8144  ax-precex 8145  ax-cnre 8146  ax-pre-ltirr 8147  ax-pre-ltwlin 8148  ax-pre-lttrn 8149  ax-pre-apti 8150  ax-pre-ltadd 8151  ax-pre-mulgt0 8152  ax-pre-mulext 8153  ax-arch 8154  ax-caucvg 8155

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5974  df-ov 6024  df-oprab 6025  df-mpo 6026  df-1st 6306  df-2nd 6307  df-recs 6474  df-frec 6560  df-pnf 8219  df-mnf 8220  df-xr 8221  df-ltxr 8222  df-le 8223  df-sub 8355  df-neg 8356  df-reap 8758  df-ap 8765  df-div 8856  df-inn 9147  df-2 9205  df-3 9206  df-4 9207  df-n0 9406  df-z 9483  df-uz 9759  df-rp 9892  df-fz 10247  df-seqfrec 10714  df-exp 10805  df-cj 11423  df-re 11424  df-im 11425  df-rsqrt 11579  df-abs 11580  df-clim 11860

This theorem is referenced by:  iserex  11920  ege2le3  12253