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Mirrors > Home > MPE Home > Th. List > psergf | Structured version Visualization version GIF version |
Description: The sequence of terms in the infinite sequence defining a power series for fixed 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) |
psergf.x | β’ (π β π β β) |
Ref | Expression |
---|---|
psergf | β’ (π β (πΊβπ):β0βΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | radcnv.a | . 2 β’ (π β π΄:β0βΆβ) | |
2 | psergf.x | . 2 β’ (π β π β β) | |
3 | pser.g | . . . . 5 β’ πΊ = (π₯ β β β¦ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) | |
4 | 3 | pserval 26359 | . . . 4 β’ (π β β β (πΊβπ) = (π β β0 β¦ ((π΄βπ) Β· (πβπ)))) |
5 | 4 | adantl 481 | . . 3 β’ ((π΄:β0βΆβ β§ π β β) β (πΊβπ) = (π β β0 β¦ ((π΄βπ) Β· (πβπ)))) |
6 | ffvelcdm 7091 | . . . . 5 β’ ((π΄:β0βΆβ β§ π β β0) β (π΄βπ) β β) | |
7 | 6 | adantlr 714 | . . . 4 β’ (((π΄:β0βΆβ β§ π β β) β§ π β β0) β (π΄βπ) β β) |
8 | expcl 14077 | . . . . 5 β’ ((π β β β§ π β β0) β (πβπ) β β) | |
9 | 8 | adantll 713 | . . . 4 β’ (((π΄:β0βΆβ β§ π β β) β§ π β β0) β (πβπ) β β) |
10 | 7, 9 | mulcld 11265 | . . 3 β’ (((π΄:β0βΆβ β§ π β β) β§ π β β0) β ((π΄βπ) Β· (πβπ)) β β) |
11 | 5, 10 | fmpt3d 7126 | . 2 β’ ((π΄:β0βΆβ β§ π β β) β (πΊβπ):β0βΆβ) |
12 | 1, 2, 11 | syl2anc 583 | 1 β’ (π β (πΊβπ):β0βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β¦ cmpt 5231 βΆwf 6544 βcfv 6548 (class class class)co 7420 βcc 11137 Β· cmul 11144 β0cn0 12503 βcexp 14059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-exp 14060 |
This theorem is referenced by: radcnvlem1 26362 radcnvlem2 26363 radcnvlem3 26364 radcnv0 26365 radcnvlt2 26368 dvradcnv 26370 pserulm 26371 pserdvlem2 26378 |
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