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Mirrors > Home > MPE Home > Th. List > radcnvlt2 | Structured version Visualization version GIF version |
Description: If π is within the open disk of radius π centered at zero, then the infinite series converges at π. (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*, < ) |
radcnvlt.x | β’ (π β π β β) |
radcnvlt.a | β’ (π β (absβπ) < π ) |
Ref | Expression |
---|---|
radcnvlt2 | β’ (π β seq0( + , (πΊβπ)) β dom β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12894 | . 2 β’ β0 = (β€β₯β0) | |
2 | 0zd 12600 | . 2 β’ (π β 0 β β€) | |
3 | pser.g | . . . 4 β’ πΊ = (π₯ β β β¦ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) | |
4 | radcnv.a | . . . 4 β’ (π β π΄:β0βΆβ) | |
5 | radcnvlt.x | . . . 4 β’ (π β π β β) | |
6 | 3, 4, 5 | psergf 26366 | . . 3 β’ (π β (πΊβπ):β0βΆβ) |
7 | fvco3 6992 | . . 3 β’ (((πΊβπ):β0βΆβ β§ π β β0) β ((abs β (πΊβπ))βπ) = (absβ((πΊβπ)βπ))) | |
8 | 6, 7 | sylan 578 | . 2 β’ ((π β§ π β β0) β ((abs β (πΊβπ))βπ) = (absβ((πΊβπ)βπ))) |
9 | 6 | ffvelcdmda 7089 | . 2 β’ ((π β§ π β β0) β ((πΊβπ)βπ) β β) |
10 | radcnv.r | . . . 4 β’ π = sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < ) | |
11 | radcnvlt.a | . . . 4 β’ (π β (absβπ) < π ) | |
12 | id 22 | . . . . . 6 β’ (π = π β π = π) | |
13 | 2fveq3 6897 | . . . . . 6 β’ (π = π β (absβ((πΊβπ)βπ)) = (absβ((πΊβπ)βπ))) | |
14 | 12, 13 | oveq12d 7434 | . . . . 5 β’ (π = π β (π Β· (absβ((πΊβπ)βπ))) = (π Β· (absβ((πΊβπ)βπ)))) |
15 | 14 | cbvmptv 5256 | . . . 4 β’ (π β β0 β¦ (π Β· (absβ((πΊβπ)βπ)))) = (π β β0 β¦ (π Β· (absβ((πΊβπ)βπ)))) |
16 | 3, 4, 10, 5, 11, 15 | radcnvlt1 26372 | . . 3 β’ (π β (seq0( + , (π β β0 β¦ (π Β· (absβ((πΊβπ)βπ))))) β dom β β§ seq0( + , (abs β (πΊβπ))) β dom β )) |
17 | 16 | simprd 494 | . 2 β’ (π β seq0( + , (abs β (πΊβπ))) β dom β ) |
18 | 1, 2, 8, 9, 17 | abscvgcvg 15797 | 1 β’ (π β seq0( + , (πΊβπ)) β dom β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3419 class class class wbr 5143 β¦ cmpt 5226 dom cdm 5672 β ccom 5676 βΆwf 6539 βcfv 6543 (class class class)co 7416 supcsup 9463 βcc 11136 βcr 11137 0cc0 11138 + caddc 11141 Β· cmul 11143 β*cxr 11277 < clt 11278 β0cn0 12502 seqcseq 13998 βcexp 14058 abscabs 15213 β cli 15460 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 |
This theorem is referenced by: pserulm 26376 pserdvlem2 26383 abelthlem3 26388 binomcxplemcvg 43856 |
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