Theorem iserex 11650

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 10595	. . . . 5 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
2	1	eleq1d 2274	. . . 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 2298	. . . . . . . . . . 11 |- ( ph -> N e. ( ZZ>= ` M ) )
9		eluzelz 9657	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	8, 9	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ZZ )
11	10	zcnd 9496	. . . . . . . . 9 |- ( ph -> N e. CC )
12		ax-1cn 8018	. . . . . . . . 9 |- 1 e. CC
13		npcan 8281	. . . . . . . . 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 10598	. . . . . . 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 2299	. . . . . . 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 11606	. . . . . . . 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 11648	. . . . . 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 4068	. . . . 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 11591	. . . . . 6 |- Rel ~~>
27	26	releldmi 4917	. . . . 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 2299	. . . . . . 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 11606	. . . . . . . 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 4066	. . . . . 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 11649	. . . . 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 4917	. . . . 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 596	. . 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 9679	. . 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 720	1 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2176   class class class wbr 4044   dom cdm 4675   ` cfv 5271  (class class class)co 5944   CCcc 7923   1c1 7926   + caddc 7928   - cmin 8243   ZZcz 9372   ZZ>=cuz 9648   seqcseq 10592   ~~> cli 11589

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-fz 10131  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590

This theorem is referenced by:  isumsplit  11802  isumrpcl  11805  geolim2  11823  cvgratz  11843  cvgratgt0  11844  mertenslemub  11845  mertenslemi1  11846  mertenslem2  11847  mertensabs  11848  eftlcvg  11998  trilpolemisumle  15977  trilpolemeq1  15979  trilpolemlt1  15980  nconstwlpolemgt0  16003