Theorem binomcxplemcvg 38553

Description: Lemma for binomcxp 38556. The sum in binomcxplemnn0 38548 and its derivative (see the next theorem, binomcxplemdvsum 38554) converge, as long as their base J is within the disk of convergence. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)

Hypotheses

Ref	Expression
binomcxp.a	|- ( ph -> A e. RR+ )
binomcxp.b	|- ( ph -> B e. RR )
binomcxp.lt	|- ( ph -> ( abs ` B ) < ( abs ` A ) )
binomcxp.c	|- ( ph -> C e. CC )
binomcxplem.f	|- F = ( j e. NN0 |-> ( CC𝑐 j ) )
binomcxplem.s	|- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
binomcxplem.r	|- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
binomcxplem.e	|- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
binomcxplem.d	|- D = ( `' abs " ( 0 [,) R ) )

Assertion

Ref	Expression
binomcxplemcvg	|- ( ( ph /\ J e. D ) -> ( seq 0 ( + , ( S ` J ) ) e. dom ~~> /\ seq 1 ( + , ( E ` J ) ) e. dom ~~> ) )

Distinct variable groups:   k, b, ph   F, b, k   J, b, k   r, b, J   ph, j   S, r

Allowed substitution hints:   ph( r)   A( j, k, r, b)   B( j, k, r, b)   C( j, k, r, b)   D( j, k, r, b)   R( j, k, r, b)   S( j, k, b)   E( j, k, r, b)   F( j, r)   J( j)

Proof of Theorem binomcxplemcvg

Step	Hyp	Ref	Expression
1		binomcxplem.s	. . 3 |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
2		binomcxp.c	. . . . . . 7 |- ( ph -> C e. CC )
3	2	adantr 481	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> C e. CC )
4		simpr 477	. . . . . 6 |- ( ( ph /\ j e. NN0 ) -> j e. NN0 )
5	3, 4	bcccl 38538	. . . . 5 |- ( ( ph /\ j e. NN0 ) -> ( CC𝑐 j ) e. CC )
6		binomcxplem.f	. . . . 5 |- F = ( j e. NN0 |-> ( CC𝑐 j ) )
7	5, 6	fmptd 6385	. . . 4 |- ( ph -> F : NN0 --> CC )
8	7	adantr 481	. . 3 |- ( ( ph /\ J e. D ) -> F : NN0 --> CC )
9		binomcxplem.r	. . 3 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
10		binomcxplem.d	. . . . . . 7 |- D = ( `' abs " ( 0 [,) R ) )
11	10	eleq2i 2693	. . . . . 6 |- ( J e. D <-> J e. ( `' abs " ( 0 [,) R ) ) )
12		absf 14077	. . . . . . 7 |- abs : CC --> RR
13		ffn 6045	. . . . . . 7 |- ( abs : CC --> RR -> abs Fn CC )
14		elpreima 6337	. . . . . . 7 |- ( abs Fn CC -> ( J e. ( `' abs " ( 0 [,) R ) ) <-> ( J e. CC /\ ( abs ` J ) e. ( 0 [,) R ) ) ) )
15	12, 13, 14	mp2b 10	. . . . . 6 |- ( J e. ( `' abs " ( 0 [,) R ) ) <-> ( J e. CC /\ ( abs ` J ) e. ( 0 [,) R ) ) )
16	11, 15	bitri 264	. . . . 5 |- ( J e. D <-> ( J e. CC /\ ( abs ` J ) e. ( 0 [,) R ) ) )
17	16	simplbi 476	. . . 4 |- ( J e. D -> J e. CC )
18	17	adantl 482	. . 3 |- ( ( ph /\ J e. D ) -> J e. CC )
19	16	simprbi 480	. . . . 5 |- ( J e. D -> ( abs ` J ) e. ( 0 [,) R ) )
20		0re 10040	. . . . . . 7 |- 0 e. RR
21		ssrab2 3687	. . . . . . . . . 10 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR
22		ressxr 10083	. . . . . . . . . 10 |- RR C_ RR*
23	21, 22	sstri 3612	. . . . . . . . 9 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR*
24		supxrcl 12145	. . . . . . . . 9 |- ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR* -> sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < ) e. RR* )
25	23, 24	ax-mp 5	. . . . . . . 8 |- sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < ) e. RR*
26	9, 25	eqeltri 2697	. . . . . . 7 |- R e. RR*
27		elico2 12237	. . . . . . 7 |- ( ( 0 e. RR /\ R e. RR* ) -> ( ( abs ` J ) e. ( 0 [,) R ) <-> ( ( abs ` J ) e. RR /\ 0 <_ ( abs ` J ) /\ ( abs ` J ) < R ) ) )
28	20, 26, 27	mp2an 708	. . . . . 6 |- ( ( abs ` J ) e. ( 0 [,) R ) <-> ( ( abs ` J ) e. RR /\ 0 <_ ( abs ` J ) /\ ( abs ` J ) < R ) )
29	28	simp3bi 1078	. . . . 5 |- ( ( abs ` J ) e. ( 0 [,) R ) -> ( abs ` J ) < R )
30	19, 29	syl 17	. . . 4 |- ( J e. D -> ( abs ` J ) < R )
31	30	adantl 482	. . 3 |- ( ( ph /\ J e. D ) -> ( abs ` J ) < R )
32	1, 8, 9, 18, 31	radcnvlt2 24173	. 2 |- ( ( ph /\ J e. D ) -> seq 0 ( + , ( S ` J ) ) e. dom ~~> )
33		binomcxplem.e	. . . . . . 7 |- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
34	33	a1i 11	. . . . . 6 |- ( ( ph /\ J e. CC ) -> E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) ) )
35		simplr 792	. . . . . . . . 9 |- ( ( ( ( ph /\ J e. CC ) /\ b = J ) /\ k e. NN ) -> b = J )
36	35	oveq1d 6665	. . . . . . . 8 |- ( ( ( ( ph /\ J e. CC ) /\ b = J ) /\ k e. NN ) -> ( b ^ ( k - 1 ) ) = ( J ^ ( k - 1 ) ) )
37	36	oveq2d 6666	. . . . . . 7 |- ( ( ( ( ph /\ J e. CC ) /\ b = J ) /\ k e. NN ) -> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) = ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) )
38	37	mpteq2dva 4744	. . . . . 6 |- ( ( ( ph /\ J e. CC ) /\ b = J ) -> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) ) )
39		simpr 477	. . . . . 6 |- ( ( ph /\ J e. CC ) -> J e. CC )
40		nnex 11026	. . . . . . . 8 |- NN e. _V
41	40	mptex 6486	. . . . . . 7 |- ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) ) e. _V
42	41	a1i 11	. . . . . 6 |- ( ( ph /\ J e. CC ) -> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) ) e. _V )
43	34, 38, 39, 42	fvmptd 6288	. . . . 5 |- ( ( ph /\ J e. CC ) -> ( E ` J ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) ) )
44	17, 43	sylan2 491	. . . 4 |- ( ( ph /\ J e. D ) -> ( E ` J ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) ) )
45	44	seqeq3d 12809	. . 3 |- ( ( ph /\ J e. D ) -> seq 1 ( + , ( E ` J ) ) = seq 1 ( + , ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) ) ) )
46		eqid 2622	. . . 4 |- ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) ) = ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) )
47	1, 9, 46, 8, 18, 31	dvradcnv2 38546	. . 3 |- ( ( ph /\ J e. D ) -> seq 1 ( + , ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( J ^ ( k - 1 ) ) ) ) ) e. dom ~~> )
48	45, 47	eqeltrd 2701	. 2 |- ( ( ph /\ J e. D ) -> seq 1 ( + , ( E ` J ) ) e. dom ~~> )
49	32, 48	jca 554	1 |- ( ( ph /\ J e. D ) -> ( seq 0 ( + , ( S ` J ) ) e. dom ~~> /\ seq 1 ( + , ( E ` J ) ) e. dom ~~> ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->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   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   RR+crp 11832   [,)cico 12177   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215  C_𝑐cbcc 38535

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-fac 13061  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  df-prod 14636  df-fallfac 14738  df-bcc 38536

This theorem is referenced by:  binomcxplemnotnn0  38555