Theorem binomcxplemradcnv 38551

Description: Lemma for binomcxp 38556. By binomcxplemfrat 38550 and radcnvrat 38513 the radius of convergence of power series sum_ k e. NN0 ( ( F ` k ) x. ( b ^ k ) ) is one. (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* , < )

Assertion

Ref	Expression
binomcxplemradcnv	|- ( ( ph /\ -. C e. NN0 ) -> R = 1 )

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

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

Proof of Theorem binomcxplemradcnv

Dummy variables i x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		binomcxplem.s	. . . 4 |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
2		simpl 473	. . . . . . . . 9 |- ( ( b = x /\ k e. NN0 ) -> b = x )
3	2	oveq1d 6665	. . . . . . . 8 |- ( ( b = x /\ k e. NN0 ) -> ( b ^ k ) = ( x ^ k ) )
4	3	oveq2d 6666	. . . . . . 7 |- ( ( b = x /\ k e. NN0 ) -> ( ( F ` k ) x. ( b ^ k ) ) = ( ( F ` k ) x. ( x ^ k ) ) )
5	4	mpteq2dva 4744	. . . . . 6 |- ( b = x -> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) = ( k e. NN0 |-> ( ( F ` k ) x. ( x ^ k ) ) ) )
6		fveq2 6191	. . . . . . . 8 |- ( k = y -> ( F ` k ) = ( F ` y ) )
7		oveq2 6658	. . . . . . . 8 |- ( k = y -> ( x ^ k ) = ( x ^ y ) )
8	6, 7	oveq12d 6668	. . . . . . 7 |- ( k = y -> ( ( F ` k ) x. ( x ^ k ) ) = ( ( F ` y ) x. ( x ^ y ) ) )
9	8	cbvmptv 4750	. . . . . 6 |- ( k e. NN0 |-> ( ( F ` k ) x. ( x ^ k ) ) ) = ( y e. NN0 |-> ( ( F ` y ) x. ( x ^ y ) ) )
10	5, 9	syl6eq 2672	. . . . 5 |- ( b = x -> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) = ( y e. NN0 |-> ( ( F ` y ) x. ( x ^ y ) ) ) )
11	10	cbvmptv 4750	. . . 4 |- ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) ) = ( x e. CC |-> ( y e. NN0 |-> ( ( F ` y ) x. ( x ^ y ) ) ) )
12	1, 11	eqtri 2644	. . 3 |- S = ( x e. CC |-> ( y e. NN0 |-> ( ( F ` y ) x. ( x ^ y ) ) ) )
13		binomcxp.c	. . . . . 6 |- ( ph -> C e. CC )
14	13	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ j e. NN0 ) -> C e. CC )
15		simpr 477	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ j e. NN0 ) -> j e. NN0 )
16	14, 15	bcccl 38538	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ j e. NN0 ) -> ( CC𝑐 j ) e. CC )
17		binomcxplem.f	. . . 4 |- F = ( j e. NN0 |-> ( CC𝑐 j ) )
18	16, 17	fmptd 6385	. . 3 |- ( ( ph /\ -. C e. NN0 ) -> F : NN0 --> CC )
19		binomcxplem.r	. . 3 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
20		oveq1 6657	. . . . . . 7 |- ( k = i -> ( k + 1 ) = ( i + 1 ) )
21	20	fveq2d 6195	. . . . . 6 |- ( k = i -> ( F ` ( k + 1 ) ) = ( F ` ( i + 1 ) ) )
22		fveq2 6191	. . . . . 6 |- ( k = i -> ( F ` k ) = ( F ` i ) )
23	21, 22	oveq12d 6668	. . . . 5 |- ( k = i -> ( ( F ` ( k + 1 ) ) / ( F ` k ) ) = ( ( F ` ( i + 1 ) ) / ( F ` i ) ) )
24	23	fveq2d 6195	. . . 4 |- ( k = i -> ( abs ` ( ( F ` ( k + 1 ) ) / ( F ` k ) ) ) = ( abs ` ( ( F ` ( i + 1 ) ) / ( F ` i ) ) ) )
25	24	cbvmptv 4750	. . 3 |- ( k e. NN0 |-> ( abs ` ( ( F ` ( k + 1 ) ) / ( F ` k ) ) ) ) = ( i e. NN0 |-> ( abs ` ( ( F ` ( i + 1 ) ) / ( F ` i ) ) ) )
26		nn0uz 11722	. . 3 |- NN0 = ( ZZ>= ` 0 )
27		0nn0 11307	. . . 4 |- 0 e. NN0
28	27	a1i 11	. . 3 |- ( ( ph /\ -. C e. NN0 ) -> 0 e. NN0 )
29	17	a1i 11	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) -> F = ( j e. NN0 |-> ( CC𝑐 j ) ) )
30		simpr 477	. . . . . 6 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) /\ j = i ) -> j = i )
31	30	oveq2d 6666	. . . . 5 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) /\ j = i ) -> ( CC𝑐 j ) = ( CC𝑐 i ) )
32		simpr 477	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) -> i e. NN0 )
33		ovexd 6680	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) -> ( CC𝑐 i ) e. _V )
34	29, 31, 32, 33	fvmptd 6288	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) -> ( F ` i ) = ( CC𝑐 i ) )
35		elfznn0 12433	. . . . . . 7 |- ( C e. ( 0 ... ( i - 1 ) ) -> C e. NN0 )
36	35	con3i 150	. . . . . 6 |- ( -. C e. NN0 -> -. C e. ( 0 ... ( i - 1 ) ) )
37	36	ad2antlr 763	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) -> -. C e. ( 0 ... ( i - 1 ) ) )
38	13	adantr 481	. . . . . . . 8 |- ( ( ph /\ i e. NN0 ) -> C e. CC )
39		simpr 477	. . . . . . . 8 |- ( ( ph /\ i e. NN0 ) -> i e. NN0 )
40	38, 39	bcc0 38539	. . . . . . 7 |- ( ( ph /\ i e. NN0 ) -> ( ( CC𝑐 i ) = 0 <-> C e. ( 0 ... ( i - 1 ) ) ) )
41	40	necon3abid 2830	. . . . . 6 |- ( ( ph /\ i e. NN0 ) -> ( ( CC𝑐 i ) =/= 0 <-> -. C e. ( 0 ... ( i - 1 ) ) ) )
42	41	adantlr 751	. . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) -> ( ( CC𝑐 i ) =/= 0 <-> -. C e. ( 0 ... ( i - 1 ) ) ) )
43	37, 42	mpbird 247	. . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) -> ( CC𝑐 i ) =/= 0 )
44	34, 43	eqnetrd 2861	. . 3 |- ( ( ( ph /\ -. C e. NN0 ) /\ i e. NN0 ) -> ( F ` i ) =/= 0 )
45		binomcxp.a	. . . 4 |- ( ph -> A e. RR+ )
46		binomcxp.b	. . . 4 |- ( ph -> B e. RR )
47		binomcxp.lt	. . . 4 |- ( ph -> ( abs ` B ) < ( abs ` A ) )
48	45, 46, 47, 13, 17	binomcxplemfrat 38550	. . 3 |- ( ( ph /\ -. C e. NN0 ) -> ( k e. NN0 |-> ( abs ` ( ( F ` ( k + 1 ) ) / ( F ` k ) ) ) ) ~~> 1 )
49		ax-1ne0 10005	. . . 4 |- 1 =/= 0
50	49	a1i 11	. . 3 |- ( ( ph /\ -. C e. NN0 ) -> 1 =/= 0 )
51	12, 18, 19, 25, 26, 28, 44, 48, 50	radcnvrat 38513	. 2 |- ( ( ph /\ -. C e. NN0 ) -> R = ( 1 / 1 ) )
52		1div1e1 10717	. 2 |- ( 1 / 1 ) = 1
53	51, 52	syl6eq 2672	1 |- ( ( ph /\ -. C e. NN0 ) -> R = 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` 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   / cdiv 10684   NN0cn0 11292   RR+crp 11832   ...cfz 12326   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-of 6897  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-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  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:  binomcxplemdvbinom  38552  binomcxplemnotnn0  38555