Theorem iserex 11280

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 10383	. . . . 5 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
2	1	eleq1d 2235	. . . 4 |- ( N = M -> ( seq N ( + , F ) e. dom ~~> <-> seq M ( + , F ) e. dom ~~> ) )
3	2	bicomd 140	. . 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 519	. . . . . . 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 2259	. . . . . . . . . . 11 |- ( ph -> N e. ( ZZ>= ` M ) )
9		eluzelz 9475	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	8, 9	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ZZ )
11	10	zcnd 9314	. . . . . . . . 9 |- ( ph -> N e. CC )
12		ax-1cn 7846	. . . . . . . . 9 |- 1 e. CC
13		npcan 8107	. . . . . . . . 9 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
14	11, 12, 13	sylancl 410	. . . . . . . 8 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
15	14	seqeq1d 10386	. . . . . . 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 520	. . . . . . . 8 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
18	17, 7	eleqtrrdi 2260	. . . . . . 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 281	. . . . . . 7 |- ( ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) /\ k e. Z ) -> ( F ` k ) e. CC )
21		simpr 109	. . . . . . . 8 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> seq M ( + , F ) e. dom ~~> )
22		climdm 11236	. . . . . . . 8 |- ( seq M ( + , F ) e. dom ~~> <-> seq M ( + , F ) ~~> ( ~~> ` seq M ( + , F ) ) )
23	21, 22	sylib 121	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq M ( + , F ) e. dom ~~> ) -> seq M ( + , F ) ~~> ( ~~> ` seq M ( + , F ) ) )
24	7, 18, 20, 23	clim2ser 11278	. . . . . 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 4006	. . . . 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 11221	. . . . . 6 |- Rel ~~>
27	26	releldmi 4843	. . . . 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 109	. . . . . . . 8 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
30	29, 7	eleqtrrdi 2260	. . . . . . 7 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( N - 1 ) e. Z )
31	30	adantr 274	. . . . . 6 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> ( N - 1 ) e. Z )
32		simpll 519	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> ph )
33	32, 19	sylan 281	. . . . . 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 109	. . . . . . . 8 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> seq N ( + , F ) e. dom ~~> )
36		climdm 11236	. . . . . . . 8 |- ( seq N ( + , F ) e. dom ~~> <-> seq N ( + , F ) ~~> ( ~~> ` seq N ( + , F ) ) )
37	35, 36	sylib 121	. . . . . . 7 |- ( ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) /\ seq N ( + , F ) e. dom ~~> ) -> seq N ( + , F ) ~~> ( ~~> ` seq N ( + , F ) ) )
38	34, 37	eqbrtrd 4004	. . . . . 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 11279	. . . . 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 4843	. . . . 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 586	. . 3 |- ( ( ph /\ ( N - 1 ) e. ( ZZ>= ` M ) ) -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
43	42	ex 114	. 2 |- ( ph -> ( ( N - 1 ) e. ( ZZ>= ` M ) -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) ) )
44		uzm1 9496	. . 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 708	1 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   class class class wbr 3982   dom cdm 4604   ` 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:  isumsplit  11432  isumrpcl  11435  geolim2  11453  cvgratz  11473  cvgratgt0  11474  mertenslemub  11475  mertenslemi1  11476  mertenslem2  11477  mertensabs  11478  eftlcvg  11628  trilpolemisumle  13917  trilpolemeq1  13919  trilpolemlt1  13920  nconstwlpolemgt0  13942