Theorem iserex 11890

Description: An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)

Hypotheses

Ref	Expression
clim2ser.1	|- Z = ( ZZ>= ` M )
iserex.2	|- ( ph -> N e. Z )
iserex.3	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )

Assertion

Ref	Expression
iserex	|- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )

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

Proof of Theorem iserex

Step	Hyp	Ref	Expression
1		seqeq1 10702	. . . . 5 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
2	1	eleq1d 2298	. . . 4 |- ( N = M -> ( seq N ( + , F ) e. dom ~~> <-> seq M ( + , F ) e. dom ~~> ) )
3	2	bicomd 141	. . 3 |- ( N = M -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
4	3	a1i 9	. 2 |- ( ph -> ( N = M -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) ) )
5		simpll 527	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> ph )
6		iserex.2	. . . . . . . . . . . 12 |- ( ph -> N e. Z )
7		clim2ser.1	. . . . . . . . . . . 12 |- Z = ( ZZ>= ` M )
8	6, 7	eleqtrdi 2322	. . . . . . . . . . 11 |- ( ph -> N e. ( ZZ>= ` M ) )
9		eluzelz 9755	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	8, 9	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ZZ )
11	10	zcnd 9593	. . . . . . . . 9 |- ( ph -> N e. CC )
12		ax-1cn 8115	. . . . . . . . 9 |- 1 e. CC
13		npcan 8378	. . . . . . . . 9 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
14	11, 12, 13	sylancl 413	. . . . . . . 8 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
15	14	seqeq1d 10705	. . . . . . 7 |- ( ph -> seq ( ( N - 1 ) + 1 ) ( + , F ) = seq N ( + , F ) )
16	5, 15	syl 14	. . . . . 6 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> seq ( ( N - 1 ) + 1 ) ( + , F ) = seq N ( + , F ) )
17		simplr 528	. . . . . . . 8 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
18	17, 7	eleqtrrdi 2323	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> ( N - 1 ) e. Z )
19		iserex.3	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
20	5, 19	sylan 283	. . . . . . 7 |- ( ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) /\ k e. Z ) -> ( F ` k ) e. CC )
21		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> seq M ( + , F ) e. dom ~~> )
22		climdm 11846	. . . . . . . 8 |- ( seq M ( + , F ) e. dom ~~> <-> seq M ( + , F ) ~~> ( ~~> ` seq M ( + , F ) ) )
23	21, 22	sylib 122	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> seq M ( + , F ) ~~> ( ~~> ` seq M ( + , F ) ) )
24	7, 18, 20, 23	clim2ser 11888	. . . . . 6 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> seq ( ( N - 1 ) + 1 ) ( + , F ) ~~> ( ( ~~> ` seq M ( + , F ) ) - ( seq M ( + , F ) ` ( N - 1 ) ) ) )
25	16, 24	eqbrtrrd 4110	. . . . 5 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> seq N ( + , F ) ~~> ( ( ~~> ` seq M ( + , F ) ) - ( seq M ( + , F ) ` ( N - 1 ) ) ) )
26		climrel 11831	. . . . . 6 |- Rel ~~>
27	26	releldmi 4969	. . . . 5 |- ( seq N ( + , F ) ~~> ( ( ~~> ` seq M ( + , F ) ) - ( seq M ( + , F ) ` ( N - 1 ) ) ) -> seq N ( + , F ) e. dom ~~> )
28	25, 27	syl 14	. . . 4 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> seq N ( + , F ) e. dom ~~> )
29		simpr 110	. . . . . . . 8 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
30	29, 7	eleqtrrdi 2323	. . . . . . 7 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( N - 1 ) e. Z )
31	30	adantr 276	. . . . . 6 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> ( N - 1 ) e. Z )
32		simpll 527	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> ph )
33	32, 19	sylan 283	. . . . . 6 |- ( ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) /\ k e. Z ) -> ( F ` k ) e. CC )
34	32, 15	syl 14	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> seq ( ( N - 1 ) + 1 ) ( + , F ) = seq N ( + , F ) )
35		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> seq N ( + , F ) e. dom ~~> )
36		climdm 11846	. . . . . . . 8 |- ( seq N ( + , F ) e. dom ~~> <-> seq N ( + , F ) ~~> ( ~~> ` seq N ( + , F ) ) )
37	35, 36	sylib 122	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> seq N ( + , F ) ~~> ( ~~> ` seq N ( + , F ) ) )
38	34, 37	eqbrtrd 4108	. . . . . 6 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> seq ( ( N - 1 ) + 1 ) ( + , F ) ~~> ( ~~> ` seq N ( + , F ) ) )
39	7, 31, 33, 38	clim2ser2 11889	. . . . 5 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> seq M ( + , F ) ~~> ( ( ~~> ` seq N ( + , F ) ) + ( seq M ( + , F ) ` ( N - 1 ) ) ) )
40	26	releldmi 4969	. . . . 5 |- ( seq M ( + , F ) ~~> ( ( ~~> ` seq N ( + , F ) ) + ( seq M ( + , F ) ` ( N - 1 ) ) ) -> seq M ( + , F ) e. dom ~~> )
41	39, 40	syl 14	. . . 4 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> seq M ( + , F ) e. dom ~~> )
42	28, 41	impbida 598	. . 3 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
43	42	ex 115	. 2 |- ( ph -> ( ( N - 1 ) e. ( ZZ>= ` M ) -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) ) )
44		uzm1 9777	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
45	8, 44	syl 14	. 2 |- ( ph -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
46	4, 43, 45	mpjaod 723	1 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   class class class wbr 4086   dom cdm 4723   ` cfv 5324  (class class class)co 6013   CCcc 8020   1c1 8023   + caddc 8025   - cmin 8340   ZZcz 9469   ZZ>=cuz 9745   seqcseq 10699   ~~> cli 11829

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-rp 9879  df-fz 10234  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830

This theorem is referenced by:  isumsplit  12042  isumrpcl  12045  geolim2  12063  cvgratz  12083  cvgratgt0  12084  mertenslemub  12085  mertenslemi1  12086  mertenslem2  12087  mertensabs  12088  eftlcvg  12238  trilpolemisumle  16578  trilpolemeq1  16580  trilpolemlt1  16581  nconstwlpolemgt0  16604