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Mirrors > Home > MPE Home > Th. List > radcnvlem3 | Structured version Visualization version GIF version |
Description: Lemma for radcnvlt1 26370, radcnvle 26372. If π is a point closer to zero than π and the power series converges at π, then it converges at π. (Contributed by Mario Carneiro, 31-Mar-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) |
psergf.x | β’ (π β π β β) |
radcnvlem2.y | β’ (π β π β β) |
radcnvlem2.a | β’ (π β (absβπ) < (absβπ)) |
radcnvlem2.c | β’ (π β seq0( + , (πΊβπ)) β dom β ) |
Ref | Expression |
---|---|
radcnvlem3 | β’ (π β seq0( + , (πΊβπ)) β dom β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12892 | . 2 β’ β0 = (β€β₯β0) | |
2 | 0zd 12598 | . 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 | psergf.x | . . . 4 β’ (π β π β β) | |
6 | 3, 4, 5 | psergf 26364 | . . 3 β’ (π β (πΊβπ):β0βΆβ) |
7 | fvco3 6991 | . . 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 7088 | . 2 β’ ((π β§ π β β0) β ((πΊβπ)βπ) β β) |
10 | radcnvlem2.y | . . 3 β’ (π β π β β) | |
11 | radcnvlem2.a | . . 3 β’ (π β (absβπ) < (absβπ)) | |
12 | radcnvlem2.c | . . 3 β’ (π β seq0( + , (πΊβπ)) β dom β ) | |
13 | 3, 4, 5, 10, 11, 12 | radcnvlem2 26366 | . 2 β’ (π β seq0( + , (abs β (πΊβπ))) β dom β ) |
14 | 1, 2, 8, 9, 13 | abscvgcvg 15795 | 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 class class class wbr 5143 β¦ cmpt 5226 dom cdm 5672 β ccom 5676 βΆwf 6538 βcfv 6542 (class class class)co 7415 βcc 11134 0cc0 11136 + caddc 11139 Β· cmul 11141 < clt 11276 β0cn0 12500 seqcseq 13996 βcexp 14056 abscabs 15211 β cli 15458 |
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 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-ico 13360 df-fz 13515 df-fzo 13658 df-fl 13787 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 |
This theorem is referenced by: radcnvle 26372 |
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