Theorem radcnvle 24174

Description: If X is a convergent point of the infinite series, then X is within the closed disk of radius R centered at zero. Or, by contraposition, the series diverges at any point strictly more than R from the origin. (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* , < )
radcnvle.x	|- ( ph -> X e. CC )
radcnvle.a	|- ( ph -> seq 0 ( + , ( G ` X ) ) e. dom ~~> )

Assertion

Ref	Expression
radcnvle	|- ( ph -> ( abs ` X ) <_ R )

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

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

Proof of Theorem radcnvle

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( ph /\ R < ( abs ` X ) ) -> R < ( abs ` X ) )
2		iccssxr 12256	. . . . . . . 8 |- ( 0 [,] +oo ) C_ RR*
3		pser.g	. . . . . . . . 9 |- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
4		radcnv.a	. . . . . . . . 9 |- ( ph -> A : NN0 --> CC )
5		radcnv.r	. . . . . . . . 9 |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
6	3, 4, 5	radcnvcl 24171	. . . . . . . 8 |- ( ph -> R e. ( 0 [,] +oo ) )
7	2, 6	sseldi 3601	. . . . . . 7 |- ( ph -> R e. RR* )
8	7	adantr 481	. . . . . 6 |- ( ( ph /\ R < ( abs ` X ) ) -> R e. RR* )
9		radcnvle.x	. . . . . . . 8 |- ( ph -> X e. CC )
10	9	abscld 14175	. . . . . . 7 |- ( ph -> ( abs ` X ) e. RR )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ R < ( abs ` X ) ) -> ( abs ` X ) e. RR )
12		0xr 10086	. . . . . . . . . . 11 |- 0 e. RR*
13		pnfxr 10092	. . . . . . . . . . 11 |- +oo e. RR*
14		elicc1 12219	. . . . . . . . . . 11 |- ( ( 0 e. RR* /\ +oo e. RR* ) -> ( R e. ( 0 [,] +oo ) <-> ( R e. RR* /\ 0 <_ R /\ R <_ +oo ) ) )
15	12, 13, 14	mp2an 708	. . . . . . . . . 10 |- ( R e. ( 0 [,] +oo ) <-> ( R e. RR* /\ 0 <_ R /\ R <_ +oo ) )
16	6, 15	sylib 208	. . . . . . . . 9 |- ( ph -> ( R e. RR* /\ 0 <_ R /\ R <_ +oo ) )
17	16	simp2d 1074	. . . . . . . 8 |- ( ph -> 0 <_ R )
18		ge0gtmnf 12003	. . . . . . . 8 |- ( ( R e. RR* /\ 0 <_ R ) -> -oo < R )
19	7, 17, 18	syl2anc 693	. . . . . . 7 |- ( ph -> -oo < R )
20	19	adantr 481	. . . . . 6 |- ( ( ph /\ R < ( abs ` X ) ) -> -oo < R )
21		ressxr 10083	. . . . . . . . . 10 |- RR C_ RR*
22	21, 10	sseldi 3601	. . . . . . . . 9 |- ( ph -> ( abs ` X ) e. RR* )
23	22	adantr 481	. . . . . . . 8 |- ( ( ph /\ R < ( abs ` X ) ) -> ( abs ` X ) e. RR* )
24		xrltle 11982	. . . . . . . 8 |- ( ( R e. RR* /\ ( abs ` X ) e. RR* ) -> ( R < ( abs ` X ) -> R <_ ( abs ` X ) ) )
25	8, 23, 24	syl2anc 693	. . . . . . 7 |- ( ( ph /\ R < ( abs ` X ) ) -> ( R < ( abs ` X ) -> R <_ ( abs ` X ) ) )
26	1, 25	mpd 15	. . . . . 6 |- ( ( ph /\ R < ( abs ` X ) ) -> R <_ ( abs ` X ) )
27		xrre 12000	. . . . . 6 |- ( ( ( R e. RR* /\ ( abs ` X ) e. RR ) /\ ( -oo < R /\ R <_ ( abs ` X ) ) ) -> R e. RR )
28	8, 11, 20, 26, 27	syl22anc 1327	. . . . 5 |- ( ( ph /\ R < ( abs ` X ) ) -> R e. RR )
29		avglt1 11270	. . . . 5 |- ( ( R e. RR /\ ( abs ` X ) e. RR ) -> ( R < ( abs ` X ) <-> R < ( ( R + ( abs ` X ) ) / 2 ) ) )
30	28, 11, 29	syl2anc 693	. . . 4 |- ( ( ph /\ R < ( abs ` X ) ) -> ( R < ( abs ` X ) <-> R < ( ( R + ( abs ` X ) ) / 2 ) ) )
31	1, 30	mpbid 222	. . 3 |- ( ( ph /\ R < ( abs ` X ) ) -> R < ( ( R + ( abs ` X ) ) / 2 ) )
32		ssrab2 3687	. . . . . . 7 |- { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } C_ RR
33	32, 21	sstri 3612	. . . . . 6 |- { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } C_ RR*
34	28, 11	readdcld 10069	. . . . . . . 8 |- ( ( ph /\ R < ( abs ` X ) ) -> ( R + ( abs ` X ) ) e. RR )
35	34	rehalfcld 11279	. . . . . . 7 |- ( ( ph /\ R < ( abs ` X ) ) -> ( ( R + ( abs ` X ) ) / 2 ) e. RR )
36	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ R < ( abs ` X ) ) -> A : NN0 --> CC )
37	35	recnd 10068	. . . . . . . 8 |- ( ( ph /\ R < ( abs ` X ) ) -> ( ( R + ( abs ` X ) ) / 2 ) e. CC )
38	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ R < ( abs ` X ) ) -> X e. CC )
39		0red 10041	. . . . . . . . . . 11 |- ( ( ph /\ R < ( abs ` X ) ) -> 0 e. RR )
40	17	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ R < ( abs ` X ) ) -> 0 <_ R )
41	39, 28, 35, 40, 31	lelttrd 10195	. . . . . . . . . . 11 |- ( ( ph /\ R < ( abs ` X ) ) -> 0 < ( ( R + ( abs ` X ) ) / 2 ) )
42	39, 35, 41	ltled 10185	. . . . . . . . . 10 |- ( ( ph /\ R < ( abs ` X ) ) -> 0 <_ ( ( R + ( abs ` X ) ) / 2 ) )
43	35, 42	absidd 14161	. . . . . . . . 9 |- ( ( ph /\ R < ( abs ` X ) ) -> ( abs ` ( ( R + ( abs ` X ) ) / 2 ) ) = ( ( R + ( abs ` X ) ) / 2 ) )
44		avglt2 11271	. . . . . . . . . . 11 |- ( ( R e. RR /\ ( abs ` X ) e. RR ) -> ( R < ( abs ` X ) <-> ( ( R + ( abs ` X ) ) / 2 ) < ( abs ` X ) ) )
45	28, 11, 44	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ R < ( abs ` X ) ) -> ( R < ( abs ` X ) <-> ( ( R + ( abs ` X ) ) / 2 ) < ( abs ` X ) ) )
46	1, 45	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ R < ( abs ` X ) ) -> ( ( R + ( abs ` X ) ) / 2 ) < ( abs ` X ) )
47	43, 46	eqbrtrd 4675	. . . . . . . 8 |- ( ( ph /\ R < ( abs ` X ) ) -> ( abs ` ( ( R + ( abs ` X ) ) / 2 ) ) < ( abs ` X ) )
48		radcnvle.a	. . . . . . . . 9 |- ( ph -> seq 0 ( + , ( G ` X ) ) e. dom ~~> )
49	48	adantr 481	. . . . . . . 8 |- ( ( ph /\ R < ( abs ` X ) ) -> seq 0 ( + , ( G ` X ) ) e. dom ~~> )
50	3, 36, 37, 38, 47, 49	radcnvlem3 24169	. . . . . . 7 |- ( ( ph /\ R < ( abs ` X ) ) -> seq 0 ( + , ( G ` ( ( R + ( abs ` X ) ) / 2 ) ) ) e. dom ~~> )
51		fveq2 6191	. . . . . . . . . 10 |- ( y = ( ( R + ( abs ` X ) ) / 2 ) -> ( G ` y ) = ( G ` ( ( R + ( abs ` X ) ) / 2 ) ) )
52	51	seqeq3d 12809	. . . . . . . . 9 |- ( y = ( ( R + ( abs ` X ) ) / 2 ) -> seq 0 ( + , ( G ` y ) ) = seq 0 ( + , ( G ` ( ( R + ( abs ` X ) ) / 2 ) ) ) )
53	52	eleq1d 2686	. . . . . . . 8 |- ( y = ( ( R + ( abs ` X ) ) / 2 ) -> ( seq 0 ( + , ( G ` y ) ) e. dom ~~> <-> seq 0 ( + , ( G ` ( ( R + ( abs ` X ) ) / 2 ) ) ) e. dom ~~> ) )
54		fveq2 6191	. . . . . . . . . . 11 |- ( r = y -> ( G ` r ) = ( G ` y ) )
55	54	seqeq3d 12809	. . . . . . . . . 10 |- ( r = y -> seq 0 ( + , ( G ` r ) ) = seq 0 ( + , ( G ` y ) ) )
56	55	eleq1d 2686	. . . . . . . . 9 |- ( r = y -> ( seq 0 ( + , ( G ` r ) ) e. dom ~~> <-> seq 0 ( + , ( G ` y ) ) e. dom ~~> ) )
57	56	cbvrabv 3199	. . . . . . . 8 |- { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } = { y e. RR | seq 0 ( + , ( G ` y ) ) e. dom ~~> }
58	53, 57	elrab2 3366	. . . . . . 7 |- ( ( ( R + ( abs ` X ) ) / 2 ) e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } <-> ( ( ( R + ( abs ` X ) ) / 2 ) e. RR /\ seq 0 ( + , ( G ` ( ( R + ( abs ` X ) ) / 2 ) ) ) e. dom ~~> ) )
59	35, 50, 58	sylanbrc 698	. . . . . 6 |- ( ( ph /\ R < ( abs ` X ) ) -> ( ( R + ( abs ` X ) ) / 2 ) e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } )
60		supxrub 12154	. . . . . 6 |- ( ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } C_ RR* /\ ( ( R + ( abs ` X ) ) / 2 ) e. { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } ) -> ( ( R + ( abs ` X ) ) / 2 ) <_ sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < ) )
61	33, 59, 60	sylancr 695	. . . . 5 |- ( ( ph /\ R < ( abs ` X ) ) -> ( ( R + ( abs ` X ) ) / 2 ) <_ sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < ) )
62	61, 5	syl6breqr 4695	. . . 4 |- ( ( ph /\ R < ( abs ` X ) ) -> ( ( R + ( abs ` X ) ) / 2 ) <_ R )
63	35, 28	lenltd 10183	. . . 4 |- ( ( ph /\ R < ( abs ` X ) ) -> ( ( ( R + ( abs ` X ) ) / 2 ) <_ R <-> -. R < ( ( R + ( abs ` X ) ) / 2 ) ) )
64	62, 63	mpbid 222	. . 3 |- ( ( ph /\ R < ( abs ` X ) ) -> -. R < ( ( R + ( abs ` X ) ) / 2 ) )
65	31, 64	pm2.65da 600	. 2 |- ( ph -> -. R < ( abs ` X ) )
66		xrlenlt 10103	. . 3 |- ( ( ( abs ` X ) e. RR* /\ R e. RR* ) -> ( ( abs ` X ) <_ R <-> -. R < ( abs ` X ) ) )
67	22, 7, 66	syl2anc 693	. 2 |- ( ph -> ( ( abs ` X ) <_ R <-> -. R < ( abs ` X ) ) )
68	65, 67	mpbird 247	1 |- ( ph -> ( abs ` X ) <_ R )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   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   + caddc 9939   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   2c2 11070   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:  pserdvlem2  24182  abelthlem1  24185  logtayl  24406