Theorem radcnvlt1 24172

Description: If X is within the open disk of radius R centered at zero, then the infinite series converges absolutely at X, and also converges when the series is multiplied by n. (Contributed by Mario Carneiro, 26-Feb-2015.)

Hypotheses

Ref	Expression
pser.g	|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
radcnv.a	|- ( ph -> A : NN0 --> CC )
radcnv.r	|- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
radcnvlt.x	|- ( ph -> X e. CC )
radcnvlt.a	|- ( ph -> ( abs ` X ) < R )
radcnvlt1.h	|- H = ( m e. NN0 |-> ( m x. ( abs ` ( ( G ` X ) ` m ) ) ) )

Assertion

Ref	Expression
radcnvlt1	|- ( ph -> ( seq 0 ( + , H ) e. dom ~~> /\ seq 0 ( + , ( abs o. ( G ` X ) ) ) e. dom ~~> ) )

Distinct variable groups:   m, n, x, A   m, H   ph, m   m, X   m, r, G

Allowed substitution hints:   ph( x, n, r)   A( r)   R( x, m, n, r)   G( x, n)   H( x, n, r)   X( x, n, r)

Proof of Theorem radcnvlt1

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		radcnvlt.a	. . . . 5 |- ( ph -> ( abs ` X ) < R )
2		ressxr 10083	. . . . . . 7 |- RR C_ RR*
3		radcnvlt.x	. . . . . . . 8 |- ( ph -> X e. CC )
4	3	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` X ) e. RR )
5	2, 4	sseldi 3601	. . . . . 6 |- ( ph -> ( abs ` X ) e. RR* )
6		iccssxr 12256	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
7		pser.g	. . . . . . . 8 |- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
8		radcnv.a	. . . . . . . 8 |- ( ph -> A : NN0 --> CC )
9		radcnv.r	. . . . . . . 8 |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
10	7, 8, 9	radcnvcl 24171	. . . . . . 7 |- ( ph -> R e. ( 0 [,] +oo ) )
11	6, 10	sseldi 3601	. . . . . 6 |- ( ph -> R e. RR* )
12		xrltnle 10105	. . . . . 6 |- ( ( ( abs ` X ) e. RR* /\ R e. RR* ) -> ( ( abs ` X ) < R <-> -. R <_ ( abs ` X ) ) )
13	5, 11, 12	syl2anc 693	. . . . 5 |- ( ph -> ( ( abs ` X ) < R <-> -. R <_ ( abs ` X ) ) )
14	1, 13	mpbid 222	. . . 4 |- ( ph -> -. R <_ ( abs ` X ) )
15	9	breq1i 4660	. . . . . 6 |- ( R <_ ( abs ` X ) <-> sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < ) <_ ( abs ` X ) )
16		ssrab2 3687	. . . . . . . 8 |- { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } C_ RR
17	16, 2	sstri 3612	. . . . . . 7 |- { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } C_ RR*
18		supxrleub 12156	. . . . . . 7 |- ( ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } C_ RR* /\ ( abs ` X ) e. RR* ) -> ( sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < ) <_ ( abs ` X ) <-> A. s e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } s <_ ( abs ` X ) ) )
19	17, 5, 18	sylancr 695	. . . . . 6 |- ( ph -> ( sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < ) <_ ( abs ` X ) <-> A. s e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } s <_ ( abs ` X ) ) )
20	15, 19	syl5bb 272	. . . . 5 |- ( ph -> ( R <_ ( abs ` X ) <-> A. s e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } s <_ ( abs ` X ) ) )
21		fveq2 6191	. . . . . . . 8 |- ( r = s -> ( G ` r ) = ( G ` s ) )
22	21	seqeq3d 12809	. . . . . . 7 |- ( r = s -> seq 0 ( + , ( G ` r ) ) = seq 0 ( + , ( G ` s ) ) )
23	22	eleq1d 2686	. . . . . 6 |- ( r = s -> ( seq 0 ( + , ( G ` r ) ) e. dom ~~> <-> seq 0 ( + , ( G ` s ) ) e. dom ~~> ) )
24	23	ralrab 3368	. . . . 5 |- ( A. s e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } s <_ ( abs ` X ) <-> A. s e. RR ( seq 0 ( + , ( G ` s ) ) e. dom ~~> -> s <_ ( abs ` X ) ) )
25	20, 24	syl6bb 276	. . . 4 |- ( ph -> ( R <_ ( abs ` X ) <-> A. s e. RR ( seq 0 ( + , ( G ` s ) ) e. dom ~~> -> s <_ ( abs ` X ) ) ) )
26	14, 25	mtbid 314	. . 3 |- ( ph -> -. A. s e. RR ( seq 0 ( + , ( G ` s ) ) e. dom ~~> -> s <_ ( abs ` X ) ) )
27		rexanali 2998	. . 3 |- ( E. s e. RR ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ -. s <_ ( abs ` X ) ) <-> -. A. s e. RR ( seq 0 ( + , ( G ` s ) ) e. dom ~~> -> s <_ ( abs ` X ) ) )
28	26, 27	sylibr 224	. 2 |- ( ph -> E. s e. RR ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ -. s <_ ( abs ` X ) ) )
29		ltnle 10117	. . . . . . 7 |- ( ( ( abs ` X ) e. RR /\ s e. RR ) -> ( ( abs ` X ) < s <-> -. s <_ ( abs ` X ) ) )
30	4, 29	sylan 488	. . . . . 6 |- ( ( ph /\ s e. RR ) -> ( ( abs ` X ) < s <-> -. s <_ ( abs ` X ) ) )
31	30	adantr 481	. . . . 5 |- ( ( ( ph /\ s e. RR ) /\ seq 0 ( + , ( G ` s ) ) e. dom ~~> ) -> ( ( abs ` X ) < s <-> -. s <_ ( abs ` X ) ) )
32	8	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> A : NN0 --> CC )
33	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> X e. CC )
34		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> s e. RR )
35	34	recnd 10068	. . . . . . . 8 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> s e. CC )
36		simprr 796	. . . . . . . . 9 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> ( abs ` X ) < s )
37		0red 10041	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> 0 e. RR )
38	33	abscld 14175	. . . . . . . . . . . 12 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> ( abs ` X ) e. RR )
39	33	absge0d 14183	. . . . . . . . . . . 12 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> 0 <_ ( abs ` X ) )
40	37, 38, 34, 39, 36	lelttrd 10195	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> 0 < s )
41	37, 34, 40	ltled 10185	. . . . . . . . . 10 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> 0 <_ s )
42	34, 41	absidd 14161	. . . . . . . . 9 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> ( abs ` s ) = s )
43	36, 42	breqtrrd 4681	. . . . . . . 8 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> ( abs ` X ) < ( abs ` s ) )
44		simprl 794	. . . . . . . 8 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> seq 0 ( + , ( G ` s ) ) e. dom ~~> )
45		radcnvlt1.h	. . . . . . . 8 |- H = ( m e. NN0 |-> ( m x. ( abs ` ( ( G ` X ) ` m ) ) ) )
46	7, 32, 33, 35, 43, 44, 45	radcnvlem1 24167	. . . . . . 7 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> seq 0 ( + , H ) e. dom ~~> )
47	7, 32, 33, 35, 43, 44	radcnvlem2 24168	. . . . . . 7 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> seq 0 ( + , ( abs o. ( G ` X ) ) ) e. dom ~~> )
48	46, 47	jca 554	. . . . . 6 |- ( ( ( ph /\ s e. RR ) /\ ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ ( abs ` X ) < s ) ) -> ( seq 0 ( + , H ) e. dom ~~> /\ seq 0 ( + , ( abs o. ( G ` X ) ) ) e. dom ~~> ) )
49	48	expr 643	. . . . 5 |- ( ( ( ph /\ s e. RR ) /\ seq 0 ( + , ( G ` s ) ) e. dom ~~> ) -> ( ( abs ` X ) < s -> ( seq 0 ( + , H ) e. dom ~~> /\ seq 0 ( + , ( abs o. ( G ` X ) ) ) e. dom ~~> ) ) )
50	31, 49	sylbird 250	. . . 4 |- ( ( ( ph /\ s e. RR ) /\ seq 0 ( + , ( G ` s ) ) e. dom ~~> ) -> ( -. s <_ ( abs ` X ) -> ( seq 0 ( + , H ) e. dom ~~> /\ seq 0 ( + , ( abs o. ( G ` X ) ) ) e. dom ~~> ) ) )
51	50	expimpd 629	. . 3 |- ( ( ph /\ s e. RR ) -> ( ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ -. s <_ ( abs ` X ) ) -> ( seq 0 ( + , H ) e. dom ~~> /\ seq 0 ( + , ( abs o. ( G ` X ) ) ) e. dom ~~> ) ) )
52	51	rexlimdva 3031	. 2 |- ( ph -> ( E. s e. RR ( seq 0 ( + , ( G ` s ) ) e. dom ~~> /\ -. s <_ ( abs ` X ) ) -> ( seq 0 ( + , H ) e. dom ~~> /\ seq 0 ( + , ( abs o. ( G ` X ) ) ) e. dom ~~> ) ) )
53	28, 52	mpd 15	1 |- ( ph -> ( seq 0 ( + , H ) e. dom ~~> /\ seq 0 ( + , ( abs o. ( G ` X ) ) ) e. dom ~~> ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   NN0cn0 11292   [,]cicc 12178   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by:  radcnvlt2  24173  dvradcnv  24175  pserulm  24176