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Mirrors > Home > MPE Home > Th. List > radcnvcl | Structured version Visualization version GIF version |
Description: The radius of convergence π of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
pser.g | β’ πΊ = (π₯ β β β¦ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) |
radcnv.a | β’ (π β π΄:β0βΆβ) |
radcnv.r | β’ π = sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < ) |
Ref | Expression |
---|---|
radcnvcl | β’ (π β π β (0[,]+β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | radcnv.r | . . 3 β’ π = sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < ) | |
2 | ssrab2 4042 | . . . . 5 β’ {π β β β£ seq0( + , (πΊβπ)) β dom β } β β | |
3 | ressxr 11206 | . . . . 5 β’ β β β* | |
4 | 2, 3 | sstri 3958 | . . . 4 β’ {π β β β£ seq0( + , (πΊβπ)) β dom β } β β* |
5 | supxrcl 13241 | . . . 4 β’ ({π β β β£ seq0( + , (πΊβπ)) β dom β } β β* β sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < ) β β*) | |
6 | 4, 5 | mp1i 13 | . . 3 β’ (π β sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < ) β β*) |
7 | 1, 6 | eqeltrid 2842 | . 2 β’ (π β π β β*) |
8 | pser.g | . . . . 5 β’ πΊ = (π₯ β β β¦ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) | |
9 | radcnv.a | . . . . 5 β’ (π β π΄:β0βΆβ) | |
10 | 8, 9 | radcnv0 25791 | . . . 4 β’ (π β 0 β {π β β β£ seq0( + , (πΊβπ)) β dom β }) |
11 | supxrub 13250 | . . . 4 β’ (({π β β β£ seq0( + , (πΊβπ)) β dom β } β β* β§ 0 β {π β β β£ seq0( + , (πΊβπ)) β dom β }) β 0 β€ sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < )) | |
12 | 4, 10, 11 | sylancr 588 | . . 3 β’ (π β 0 β€ sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < )) |
13 | 12, 1 | breqtrrdi 5152 | . 2 β’ (π β 0 β€ π ) |
14 | pnfge 13058 | . . 3 β’ (π β β* β π β€ +β) | |
15 | 7, 14 | syl 17 | . 2 β’ (π β π β€ +β) |
16 | 0xr 11209 | . . 3 β’ 0 β β* | |
17 | pnfxr 11216 | . . 3 β’ +β β β* | |
18 | elicc1 13315 | . . 3 β’ ((0 β β* β§ +β β β*) β (π β (0[,]+β) β (π β β* β§ 0 β€ π β§ π β€ +β))) | |
19 | 16, 17, 18 | mp2an 691 | . 2 β’ (π β (0[,]+β) β (π β β* β§ 0 β€ π β§ π β€ +β)) |
20 | 7, 13, 15, 19 | syl3anbrc 1344 | 1 β’ (π β π β (0[,]+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1088 = wceq 1542 β wcel 2107 {crab 3410 β wss 3915 class class class wbr 5110 β¦ cmpt 5193 dom cdm 5638 βΆwf 6497 βcfv 6501 (class class class)co 7362 supcsup 9383 βcc 11056 βcr 11057 0cc0 11058 + caddc 11061 Β· cmul 11063 +βcpnf 11193 β*cxr 11195 < clt 11196 β€ cle 11197 β0cn0 12420 [,]cicc 13274 seqcseq 13913 βcexp 13974 β cli 15373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-icc 13278 df-fz 13432 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 |
This theorem is referenced by: radcnvlt1 25793 radcnvle 25795 pserulm 25797 psercnlem2 25799 psercnlem1 25800 psercn 25801 pserdvlem1 25802 pserdvlem2 25803 abelthlem3 25808 abelth 25816 logtayl 26031 |
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