Theorem clim2ser 12084

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 2238	. 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 2331	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		peano2uz 9965	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
6	4, 5	syl 14	. . 3 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
7		eluzelz 9913	. . 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 9908	. . . . 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 10901	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
14	13, 2	ffvelcdmd 5838	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) e. CC )
15		seqex 10867	. . 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 2332	. . . 4 |- ( ph -> ( N + 1 ) e. Z )
19	3	uztrn2 9922	. . . 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 5838	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) e. CC )
22		addcl 8297	. . . . . 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 8302	. . . . . 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 2305	. . . . . . 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 481	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
31	23, 25, 26, 27, 30	seq3split 10906	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) = ( ( seq M ( + , F ) ` N ) + ( seq ( N + 1 ) ( + , F ) ` j ) ) )
32	31	oveq1d 6093	. . 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 9922	. . . . . . . 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 10901	. . . . 5 |- ( ph -> seq ( N + 1 ) ( + , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
38	37	ffvelcdmda 5837	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( + , F ) ` j ) e. CC )
39	33, 38	pncan2d 8632	. . 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 2272	. 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 12079	1 |- ( ph -> seq ( N + 1 ) ( + , F ) ~~> ( A - ( seq M ( + , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4128   -->wf 5371   ` cfv 5375  (class class class)co 6078   CCcc 8170   1c1 8173   + caddc 8175   - cmin 8490   ZZcz 9626   ZZ>=cuz 9903   seqcseq 10865   ~~> cli 12025

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290  ax-arch 8291  ax-caucvg 8292

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-frec 6655  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-n0 9546  df-z 9627  df-uz 9904  df-rp 10037  df-fz 10394  df-seqfrec 10866  df-exp 10957  df-cj 11588  df-re 11589  df-im 11590  df-rsqrt 11745  df-abs 11746  df-clim 12026

This theorem is referenced by:  iserex  12086  ege2le3  12419