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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 4072 | . . . . 5 β’ {π β β β£ seq0( + , (πΊβπ)) β dom β } β β | |
3 | ressxr 11262 | . . . . 5 β’ β β β* | |
4 | 2, 3 | sstri 3986 | . . . 4 β’ {π β β β£ seq0( + , (πΊβπ)) β dom β } β β* |
5 | supxrcl 13300 | . . . 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 2831 | . 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 26307 | . . . 4 β’ (π β 0 β {π β β β£ seq0( + , (πΊβπ)) β dom β }) |
11 | supxrub 13309 | . . . 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 586 | . . 3 β’ (π β 0 β€ sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < )) |
13 | 12, 1 | breqtrrdi 5183 | . 2 β’ (π β 0 β€ π ) |
14 | pnfge 13116 | . . 3 β’ (π β β* β π β€ +β) | |
15 | 7, 14 | syl 17 | . 2 β’ (π β π β€ +β) |
16 | 0xr 11265 | . . 3 β’ 0 β β* | |
17 | pnfxr 11272 | . . 3 β’ +β β β* | |
18 | elicc1 13374 | . . 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 689 | . 2 β’ (π β (0[,]+β) β (π β β* β§ 0 β€ π β§ π β€ +β)) |
20 | 7, 13, 15, 19 | syl3anbrc 1340 | 1 β’ (π β π β (0[,]+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3426 β wss 3943 class class class wbr 5141 β¦ cmpt 5224 dom cdm 5669 βΆwf 6533 βcfv 6537 (class class class)co 7405 supcsup 9437 βcc 11110 βcr 11111 0cc0 11112 + caddc 11115 Β· cmul 11117 +βcpnf 11249 β*cxr 11251 < clt 11252 β€ cle 11253 β0cn0 12476 [,]cicc 13333 seqcseq 13972 βcexp 14032 β cli 15434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-icc 13337 df-fz 13491 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 |
This theorem is referenced by: radcnvlt1 26309 radcnvle 26311 pserulm 26313 psercnlem2 26316 psercnlem1 26317 psercn 26318 pserdvlem1 26319 pserdvlem2 26320 abelthlem3 26325 abelth 26333 logtayl 26549 |
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