Theorem iserex 11349

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 10450	. . . . 5 |- ( N = M -> seq N ( + , F ) = seq M ( + , F ) )
2	1	eleq1d 2246	. . . 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 2270	. . . . . . . . . . 11 |- ( ph -> N e. ( ZZ>= ` M ) )
9		eluzelz 9539	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
10	8, 9	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ZZ )
11	10	zcnd 9378	. . . . . . . . 9 |- ( ph -> N e. CC )
12		ax-1cn 7906	. . . . . . . . 9 |- 1 e. CC
13		npcan 8168	. . . . . . . . 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 10453	. . . . . . 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 2271	. . . . . . 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 11305	. . . . . . . 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 11347	. . . . . 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 4029	. . . . 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 11290	. . . . . 6 |- Rel ~~>
27	26	releldmi 4868	. . . . 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 2271	. . . . . . 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 11305	. . . . . . . 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 4027	. . . . . 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 11348	. . . . 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 4868	. . . . 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 9560	. . 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 718	1 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   = wceq 1353   e. wcel 2148   class class class wbr 4005   dom cdm 4628   ` cfv 5218  (class class class)co 5877   CCcc 7811   1c1 7814   + caddc 7816   - cmin 8130   ZZcz 9255   ZZ>=cuz 9530   seqcseq 10447   ~~> cli 11288

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-rp 9656  df-fz 10011  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289

This theorem is referenced by:  isumsplit  11501  isumrpcl  11504  geolim2  11522  cvgratz  11542  cvgratgt0  11543  mertenslemub  11544  mertenslemi1  11545  mertenslem2  11546  mertensabs  11547  eftlcvg  11697  trilpolemisumle  14871  trilpolemeq1  14873  trilpolemlt1  14874  nconstwlpolemgt0  14897