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Mirrors > Home > MPE Home > Th. List > radcnvlem3 | Structured version Visualization version GIF version |
Description: Lemma for radcnvlt1 25793, radcnvle 25795. 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 12812 | . 2 β’ β0 = (β€β₯β0) | |
2 | 0zd 12518 | . 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 25787 | . . 3 β’ (π β (πΊβπ):β0βΆβ) |
7 | fvco3 6945 | . . 3 β’ (((πΊβπ):β0βΆβ β§ π β β0) β ((abs β (πΊβπ))βπ) = (absβ((πΊβπ)βπ))) | |
8 | 6, 7 | sylan 581 | . 2 β’ ((π β§ π β β0) β ((abs β (πΊβπ))βπ) = (absβ((πΊβπ)βπ))) |
9 | 6 | ffvelcdmda 7040 | . 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 25789 | . 2 β’ (π β seq0( + , (abs β (πΊβπ))) β dom β ) |
14 | 1, 2, 8, 9, 13 | abscvgcvg 15711 | 1 β’ (π β seq0( + , (πΊβπ)) β dom β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 class class class wbr 5110 β¦ cmpt 5193 dom cdm 5638 β ccom 5642 βΆwf 6497 βcfv 6501 (class class class)co 7362 βcc 11056 0cc0 11058 + caddc 11061 Β· cmul 11063 < clt 11196 β0cn0 12420 seqcseq 13913 βcexp 13974 abscabs 15126 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-int 4913 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-se 5594 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-isom 6510 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-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 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-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-ico 13277 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 |
This theorem is referenced by: radcnvle 25795 |
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