Theorem dvradcnv2 38546

Description: The radius of convergence of the (formal) derivative H of the power series G is (at least) as large as the radius of convergence of G. This version of dvradcnv 24175 uses a shifted version of H to match the sum form of ( CC _D F ) in pserdv2 24184 (and shows how to use uzmptshftfval 38545 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.)

Hypotheses

Ref	Expression
dvradcnv2.g	|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
dvradcnv2.r	|- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
dvradcnv2.h	|- H = ( n e. NN |-> ( ( n x. ( A ` n ) ) x. ( X ^ ( n - 1 ) ) ) )
dvradcnv2.a	|- ( ph -> A : NN0 --> CC )
dvradcnv2.x	|- ( ph -> X e. CC )
dvradcnv2.l	|- ( ph -> ( abs ` X ) < R )

Assertion

Ref	Expression
dvradcnv2	|- ( ph -> seq 1 ( + , H ) e. dom ~~> )

Distinct variable groups:   x, r, X   x, n, A   n, X   G, r

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

Proof of Theorem dvradcnv2

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 10032	. . . . 5 |- 0 e. CC
2		ax-1cn 9994	. . . . 5 |- 1 e. CC
3	1, 2	subnegi 10360	. . . 4 |- ( 0 - -u 1 ) = ( 0 + 1 )
4		0p1e1 11132	. . . 4 |- ( 0 + 1 ) = 1
5	3, 4	eqtri 2644	. . 3 |- ( 0 - -u 1 ) = 1
6		seqeq1 12804	. . 3 |- ( ( 0 - -u 1 ) = 1 -> seq ( 0 - -u 1 ) ( + , H ) = seq 1 ( + , H ) )
7	5, 6	ax-mp 5	. 2 |- seq ( 0 - -u 1 ) ( + , H ) = seq 1 ( + , H )
8		dvradcnv2.h	. . . . . . . 8 |- H = ( n e. NN |-> ( ( n x. ( A ` n ) ) x. ( X ^ ( n - 1 ) ) ) )
9		ovex 6678	. . . . . . . 8 |- ( ( n x. ( A ` n ) ) x. ( X ^ ( n - 1 ) ) ) e. _V
10		id 22	. . . . . . . . . 10 |- ( n = ( m - -u 1 ) -> n = ( m - -u 1 ) )
11		fveq2 6191	. . . . . . . . . 10 |- ( n = ( m - -u 1 ) -> ( A ` n ) = ( A ` ( m - -u 1 ) ) )
12	10, 11	oveq12d 6668	. . . . . . . . 9 |- ( n = ( m - -u 1 ) -> ( n x. ( A ` n ) ) = ( ( m - -u 1 ) x. ( A ` ( m - -u 1 ) ) ) )
13		oveq1 6657	. . . . . . . . . 10 |- ( n = ( m - -u 1 ) -> ( n - 1 ) = ( ( m - -u 1 ) - 1 ) )
14	13	oveq2d 6666	. . . . . . . . 9 |- ( n = ( m - -u 1 ) -> ( X ^ ( n - 1 ) ) = ( X ^ ( ( m - -u 1 ) - 1 ) ) )
15	12, 14	oveq12d 6668	. . . . . . . 8 |- ( n = ( m - -u 1 ) -> ( ( n x. ( A ` n ) ) x. ( X ^ ( n - 1 ) ) ) = ( ( ( m - -u 1 ) x. ( A ` ( m - -u 1 ) ) ) x. ( X ^ ( ( m - -u 1 ) - 1 ) ) ) )
16		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
17		nn0uz 11722	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
18		1pneg1e0 11129	. . . . . . . . . 10 |- ( 1 + -u 1 ) = 0
19	18	fveq2i 6194	. . . . . . . . 9 |- ( ZZ>= ` ( 1 + -u 1 ) ) = ( ZZ>= ` 0 )
20	17, 19	eqtr4i 2647	. . . . . . . 8 |- NN0 = ( ZZ>= ` ( 1 + -u 1 ) )
21		1zzd 11408	. . . . . . . 8 |- ( ph -> 1 e. ZZ )
22	21	znegcld 11484	. . . . . . . 8 |- ( ph -> -u 1 e. ZZ )
23	8, 9, 15, 16, 20, 21, 22	uzmptshftfval 38545	. . . . . . 7 |- ( ph -> ( H shift -u 1 ) = ( m e. NN0 |-> ( ( ( m - -u 1 ) x. ( A ` ( m - -u 1 ) ) ) x. ( X ^ ( ( m - -u 1 ) - 1 ) ) ) ) )
24		nn0cn 11302	. . . . . . . . . . . 12 |- ( m e. NN0 -> m e. CC )
25	24	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> m e. CC )
26		1cnd 10056	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> 1 e. CC )
27	25, 26	subnegd 10399	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( m - -u 1 ) = ( m + 1 ) )
28	27	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( A ` ( m - -u 1 ) ) = ( A ` ( m + 1 ) ) )
29	27, 28	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( m - -u 1 ) x. ( A ` ( m - -u 1 ) ) ) = ( ( m + 1 ) x. ( A ` ( m + 1 ) ) ) )
30	27	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> ( ( m - -u 1 ) - 1 ) = ( ( m + 1 ) - 1 ) )
31	25, 26	pncand 10393	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> ( ( m + 1 ) - 1 ) = m )
32	30, 31	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( ( m - -u 1 ) - 1 ) = m )
33	32	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( X ^ ( ( m - -u 1 ) - 1 ) ) = ( X ^ m ) )
34	29, 33	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ m e. NN0 ) -> ( ( ( m - -u 1 ) x. ( A ` ( m - -u 1 ) ) ) x. ( X ^ ( ( m - -u 1 ) - 1 ) ) ) = ( ( ( m + 1 ) x. ( A ` ( m + 1 ) ) ) x. ( X ^ m ) ) )
35	34	mpteq2dva 4744	. . . . . . 7 |- ( ph -> ( m e. NN0 |-> ( ( ( m - -u 1 ) x. ( A ` ( m - -u 1 ) ) ) x. ( X ^ ( ( m - -u 1 ) - 1 ) ) ) ) = ( m e. NN0 |-> ( ( ( m + 1 ) x. ( A ` ( m + 1 ) ) ) x. ( X ^ m ) ) ) )
36	23, 35	eqtrd 2656	. . . . . 6 |- ( ph -> ( H shift -u 1 ) = ( m e. NN0 |-> ( ( ( m + 1 ) x. ( A ` ( m + 1 ) ) ) x. ( X ^ m ) ) ) )
37	36	seqeq3d 12809	. . . . 5 |- ( ph -> seq 0 ( + , ( H shift -u 1 ) ) = seq 0 ( + , ( m e. NN0 |-> ( ( ( m + 1 ) x. ( A ` ( m + 1 ) ) ) x. ( X ^ m ) ) ) ) )
38		dvradcnv2.g	. . . . . . 7 |- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
39		fveq2 6191	. . . . . . . . . 10 |- ( n = m -> ( A ` n ) = ( A ` m ) )
40		oveq2 6658	. . . . . . . . . 10 |- ( n = m -> ( x ^ n ) = ( x ^ m ) )
41	39, 40	oveq12d 6668	. . . . . . . . 9 |- ( n = m -> ( ( A ` n ) x. ( x ^ n ) ) = ( ( A ` m ) x. ( x ^ m ) ) )
42	41	cbvmptv 4750	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) = ( m e. NN0 |-> ( ( A ` m ) x. ( x ^ m ) ) )
43	42	mpteq2i 4741	. . . . . . 7 |- ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) ) = ( x e. CC |-> ( m e. NN0 |-> ( ( A ` m ) x. ( x ^ m ) ) ) )
44	38, 43	eqtri 2644	. . . . . 6 |- G = ( x e. CC |-> ( m e. NN0 |-> ( ( A ` m ) x. ( x ^ m ) ) ) )
45		dvradcnv2.r	. . . . . 6 |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
46		eqid 2622	. . . . . 6 |- ( m e. NN0 |-> ( ( ( m + 1 ) x. ( A ` ( m + 1 ) ) ) x. ( X ^ m ) ) ) = ( m e. NN0 |-> ( ( ( m + 1 ) x. ( A ` ( m + 1 ) ) ) x. ( X ^ m ) ) )
47		dvradcnv2.a	. . . . . 6 |- ( ph -> A : NN0 --> CC )
48		dvradcnv2.x	. . . . . 6 |- ( ph -> X e. CC )
49		dvradcnv2.l	. . . . . 6 |- ( ph -> ( abs ` X ) < R )
50	44, 45, 46, 47, 48, 49	dvradcnv 24175	. . . . 5 |- ( ph -> seq 0 ( + , ( m e. NN0 |-> ( ( ( m + 1 ) x. ( A ` ( m + 1 ) ) ) x. ( X ^ m ) ) ) ) e. dom ~~> )
51	37, 50	eqeltrd 2701	. . . 4 |- ( ph -> seq 0 ( + , ( H shift -u 1 ) ) e. dom ~~> )
52		climdm 14285	. . . 4 |- ( seq 0 ( + , ( H shift -u 1 ) ) e. dom ~~> <-> seq 0 ( + , ( H shift -u 1 ) ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) )
53	51, 52	sylib 208	. . 3 |- ( ph -> seq 0 ( + , ( H shift -u 1 ) ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) )
54		0z 11388	. . . . . . 7 |- 0 e. ZZ
55		neg1z 11413	. . . . . . 7 |- -u 1 e. ZZ
56		nnex 11026	. . . . . . . . . 10 |- NN e. _V
57	56	mptex 6486	. . . . . . . . 9 |- ( n e. NN |-> ( ( n x. ( A ` n ) ) x. ( X ^ ( n - 1 ) ) ) ) e. _V
58	8, 57	eqeltri 2697	. . . . . . . 8 |- H e. _V
59	58	seqshft 13825	. . . . . . 7 |- ( ( 0 e. ZZ /\ -u 1 e. ZZ ) -> seq 0 ( + , ( H shift -u 1 ) ) = ( seq ( 0 - -u 1 ) ( + , H ) shift -u 1 ) )
60	54, 55, 59	mp2an 708	. . . . . 6 |- seq 0 ( + , ( H shift -u 1 ) ) = ( seq ( 0 - -u 1 ) ( + , H ) shift -u 1 )
61	60	breq1i 4660	. . . . 5 |- ( seq 0 ( + , ( H shift -u 1 ) ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) <-> ( seq ( 0 - -u 1 ) ( + , H ) shift -u 1 ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) )
62		seqex 12803	. . . . . 6 |- seq ( 0 - -u 1 ) ( + , H ) e. _V
63		climshft 14307	. . . . . 6 |- ( ( -u 1 e. ZZ /\ seq ( 0 - -u 1 ) ( + , H ) e. _V ) -> ( ( seq ( 0 - -u 1 ) ( + , H ) shift -u 1 ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) <-> seq ( 0 - -u 1 ) ( + , H ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) ) )
64	55, 62, 63	mp2an 708	. . . . 5 |- ( ( seq ( 0 - -u 1 ) ( + , H ) shift -u 1 ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) <-> seq ( 0 - -u 1 ) ( + , H ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) )
65	61, 64	bitri 264	. . . 4 |- ( seq 0 ( + , ( H shift -u 1 ) ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) <-> seq ( 0 - -u 1 ) ( + , H ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) )
66		fvex 6201	. . . . 5 |- ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) e. _V
67	62, 66	breldm 5329	. . . 4 |- ( seq ( 0 - -u 1 ) ( + , H ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) -> seq ( 0 - -u 1 ) ( + , H ) e. dom ~~> )
68	65, 67	sylbi 207	. . 3 |- ( seq 0 ( + , ( H shift -u 1 ) ) ~~> ( ~~> ` seq 0 ( + , ( H shift -u 1 ) ) ) -> seq ( 0 - -u 1 ) ( + , H ) e. dom ~~> )
69	53, 68	syl 17	. 2 |- ( ph -> seq ( 0 - -u 1 ) ( + , H ) e. dom ~~> )
70	7, 69	syl5eqelr 2706	1 |- ( ph -> seq 1 ( + , H ) e. dom ~~> )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ^cexp 12860   shift cshi 13806   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-shft 13807  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:  binomcxplemcvg  38553