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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stirlingr | Structured version Visualization version GIF version |
Description: Stirling's approximation formula for π factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 44262 is proven for convergence in the topology of complex numbers. The variable π is used to denote convergence with respect to the standard topology on the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
stirlingr.1 | β’ π = (π β β0 β¦ ((ββ((2 Β· Ο) Β· π)) Β· ((π / e)βπ))) |
stirlingr.2 | β’ π = (βπ‘β(topGenβran (,))) |
Ref | Expression |
---|---|
stirlingr | β’ (π β β β¦ ((!βπ) / (πβπ)))π 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stirlingr.1 | . . 3 β’ π = (π β β0 β¦ ((ββ((2 Β· Ο) Β· π)) Β· ((π / e)βπ))) | |
2 | 1 | stirling 44262 | . 2 β’ (π β β β¦ ((!βπ) / (πβπ))) β 1 |
3 | stirlingr.2 | . . . 4 β’ π = (βπ‘β(topGenβran (,))) | |
4 | nnuz 12798 | . . . 4 β’ β = (β€β₯β1) | |
5 | 1zzd 12530 | . . . 4 β’ (β€ β 1 β β€) | |
6 | eqid 2736 | . . . . . 6 β’ (π β β β¦ ((!βπ) / (πβπ))) = (π β β β¦ ((!βπ) / (πβπ))) | |
7 | nnnn0 12416 | . . . . . . . 8 β’ (π β β β π β β0) | |
8 | faccl 14175 | . . . . . . . 8 β’ (π β β0 β (!βπ) β β) | |
9 | nnre 12156 | . . . . . . . 8 β’ ((!βπ) β β β (!βπ) β β) | |
10 | 7, 8, 9 | 3syl 18 | . . . . . . 7 β’ (π β β β (!βπ) β β) |
11 | 2re 12223 | . . . . . . . . . . . . . 14 β’ 2 β β | |
12 | 11 | a1i 11 | . . . . . . . . . . . . 13 β’ (π β β β 2 β β) |
13 | pire 25799 | . . . . . . . . . . . . . 14 β’ Ο β β | |
14 | 13 | a1i 11 | . . . . . . . . . . . . 13 β’ (π β β β Ο β β) |
15 | 12, 14 | remulcld 11181 | . . . . . . . . . . . 12 β’ (π β β β (2 Β· Ο) β β) |
16 | nnre 12156 | . . . . . . . . . . . 12 β’ (π β β β π β β) | |
17 | 15, 16 | remulcld 11181 | . . . . . . . . . . 11 β’ (π β β β ((2 Β· Ο) Β· π) β β) |
18 | 0re 11153 | . . . . . . . . . . . . . . 15 β’ 0 β β | |
19 | 18 | a1i 11 | . . . . . . . . . . . . . 14 β’ (π β β β 0 β β) |
20 | 2pos 12252 | . . . . . . . . . . . . . . 15 β’ 0 < 2 | |
21 | 20 | a1i 11 | . . . . . . . . . . . . . 14 β’ (π β β β 0 < 2) |
22 | 19, 12, 21 | ltled 11299 | . . . . . . . . . . . . 13 β’ (π β β β 0 β€ 2) |
23 | pipos 25801 | . . . . . . . . . . . . . . 15 β’ 0 < Ο | |
24 | 18, 13, 23 | ltleii 11274 | . . . . . . . . . . . . . 14 β’ 0 β€ Ο |
25 | 24 | a1i 11 | . . . . . . . . . . . . 13 β’ (π β β β 0 β€ Ο) |
26 | 12, 14, 22, 25 | mulge0d 11728 | . . . . . . . . . . . 12 β’ (π β β β 0 β€ (2 Β· Ο)) |
27 | 7 | nn0ge0d 12472 | . . . . . . . . . . . 12 β’ (π β β β 0 β€ π) |
28 | 15, 16, 26, 27 | mulge0d 11728 | . . . . . . . . . . 11 β’ (π β β β 0 β€ ((2 Β· Ο) Β· π)) |
29 | 17, 28 | resqrtcld 15294 | . . . . . . . . . 10 β’ (π β β β (ββ((2 Β· Ο) Β· π)) β β) |
30 | ere 15963 | . . . . . . . . . . . . 13 β’ e β β | |
31 | 30 | a1i 11 | . . . . . . . . . . . 12 β’ (π β β β e β β) |
32 | epos 16081 | . . . . . . . . . . . . . 14 β’ 0 < e | |
33 | 18, 32 | gtneii 11263 | . . . . . . . . . . . . 13 β’ e β 0 |
34 | 33 | a1i 11 | . . . . . . . . . . . 12 β’ (π β β β e β 0) |
35 | 16, 31, 34 | redivcld 11979 | . . . . . . . . . . 11 β’ (π β β β (π / e) β β) |
36 | 35, 7 | reexpcld 14060 | . . . . . . . . . 10 β’ (π β β β ((π / e)βπ) β β) |
37 | 29, 36 | remulcld 11181 | . . . . . . . . 9 β’ (π β β β ((ββ((2 Β· Ο) Β· π)) Β· ((π / e)βπ)) β β) |
38 | 1 | fvmpt2 6956 | . . . . . . . . 9 β’ ((π β β0 β§ ((ββ((2 Β· Ο) Β· π)) Β· ((π / e)βπ)) β β) β (πβπ) = ((ββ((2 Β· Ο) Β· π)) Β· ((π / e)βπ))) |
39 | 7, 37, 38 | syl2anc 584 | . . . . . . . 8 β’ (π β β β (πβπ) = ((ββ((2 Β· Ο) Β· π)) Β· ((π / e)βπ))) |
40 | 2rp 12912 | . . . . . . . . . . . . 13 β’ 2 β β+ | |
41 | 40 | a1i 11 | . . . . . . . . . . . 12 β’ (π β β β 2 β β+) |
42 | pirp 25802 | . . . . . . . . . . . . 13 β’ Ο β β+ | |
43 | 42 | a1i 11 | . . . . . . . . . . . 12 β’ (π β β β Ο β β+) |
44 | 41, 43 | rpmulcld 12965 | . . . . . . . . . . 11 β’ (π β β β (2 Β· Ο) β β+) |
45 | nnrp 12918 | . . . . . . . . . . 11 β’ (π β β β π β β+) | |
46 | 44, 45 | rpmulcld 12965 | . . . . . . . . . 10 β’ (π β β β ((2 Β· Ο) Β· π) β β+) |
47 | 46 | rpsqrtcld 15288 | . . . . . . . . 9 β’ (π β β β (ββ((2 Β· Ο) Β· π)) β β+) |
48 | epr 16082 | . . . . . . . . . . . 12 β’ e β β+ | |
49 | 48 | a1i 11 | . . . . . . . . . . 11 β’ (π β β β e β β+) |
50 | 45, 49 | rpdivcld 12966 | . . . . . . . . . 10 β’ (π β β β (π / e) β β+) |
51 | nnz 12516 | . . . . . . . . . 10 β’ (π β β β π β β€) | |
52 | 50, 51 | rpexpcld 14142 | . . . . . . . . 9 β’ (π β β β ((π / e)βπ) β β+) |
53 | 47, 52 | rpmulcld 12965 | . . . . . . . 8 β’ (π β β β ((ββ((2 Β· Ο) Β· π)) Β· ((π / e)βπ)) β β+) |
54 | 39, 53 | eqeltrd 2838 | . . . . . . 7 β’ (π β β β (πβπ) β β+) |
55 | 10, 54 | rerpdivcld 12980 | . . . . . 6 β’ (π β β β ((!βπ) / (πβπ)) β β) |
56 | 6, 55 | fmpti 7056 | . . . . 5 β’ (π β β β¦ ((!βπ) / (πβπ))):ββΆβ |
57 | 56 | a1i 11 | . . . 4 β’ (β€ β (π β β β¦ ((!βπ) / (πβπ))):ββΆβ) |
58 | 3, 4, 5, 57 | climreeq 43786 | . . 3 β’ (β€ β ((π β β β¦ ((!βπ) / (πβπ)))π 1 β (π β β β¦ ((!βπ) / (πβπ))) β 1)) |
59 | 58 | mptru 1548 | . 2 β’ ((π β β β¦ ((!βπ) / (πβπ)))π 1 β (π β β β¦ ((!βπ) / (πβπ))) β 1) |
60 | 2, 59 | mpbir 230 | 1 β’ (π β β β¦ ((!βπ) / (πβπ)))π 1 |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1541 β€wtru 1542 β wcel 2106 β wne 2941 class class class wbr 5103 β¦ cmpt 5186 ran crn 5632 βΆwf 6489 βcfv 6493 (class class class)co 7353 βcr 11046 0cc0 11047 1c1 11048 Β· cmul 11052 < clt 11185 β€ cle 11186 / cdiv 11808 βcn 12149 2c2 12204 β0cn0 12409 β+crp 12907 (,)cioo 13256 βcexp 13959 !cfa 14165 βcsqrt 15110 β cli 15358 eceu 15937 Οcpi 15941 topGenctg 17311 βπ‘clm 22561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-inf2 9573 ax-cc 10367 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 ax-addf 11126 ax-mulf 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-symdif 4200 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-disj 5069 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-ofr 7614 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-2o 8409 df-oadd 8412 df-omul 8413 df-er 8644 df-map 8763 df-pm 8764 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9442 df-dju 9833 df-card 9871 df-acn 9874 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-xnn0 12482 df-z 12496 df-dec 12615 df-uz 12760 df-q 12866 df-rp 12908 df-xneg 13025 df-xadd 13026 df-xmul 13027 df-ioo 13260 df-ioc 13261 df-ico 13262 df-icc 13263 df-fz 13417 df-fzo 13560 df-fl 13689 df-mod 13767 df-seq 13899 df-exp 13960 df-fac 14166 df-bc 14195 df-hash 14223 df-shft 14944 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-limsup 15345 df-clim 15362 df-rlim 15363 df-sum 15563 df-ef 15942 df-e 15943 df-sin 15944 df-cos 15945 df-tan 15946 df-pi 15947 df-dvds 16129 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-starv 17140 df-sca 17141 df-vsca 17142 df-ip 17143 df-tset 17144 df-ple 17145 df-ds 17147 df-unif 17148 df-hom 17149 df-cco 17150 df-rest 17296 df-topn 17297 df-0g 17315 df-gsum 17316 df-topgen 17317 df-pt 17318 df-prds 17321 df-xrs 17376 df-qtop 17381 df-imas 17382 df-xps 17384 df-mre 17458 df-mrc 17459 df-acs 17461 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-submnd 18594 df-mulg 18864 df-cntz 19088 df-cmn 19555 df-psmet 20773 df-xmet 20774 df-met 20775 df-bl 20776 df-mopn 20777 df-fbas 20778 df-fg 20779 df-cnfld 20782 df-top 22227 df-topon 22244 df-topsp 22266 df-bases 22280 df-cld 22354 df-ntr 22355 df-cls 22356 df-nei 22433 df-lp 22471 df-perf 22472 df-cn 22562 df-cnp 22563 df-lm 22564 df-haus 22650 df-cmp 22722 df-tx 22897 df-hmeo 23090 df-fil 23181 df-fm 23273 df-flim 23274 df-flf 23275 df-xms 23657 df-ms 23658 df-tms 23659 df-cncf 24225 df-ovol 24812 df-vol 24813 df-mbf 24967 df-itg1 24968 df-itg2 24969 df-ibl 24970 df-itg 24971 df-0p 25018 df-limc 25214 df-dv 25215 df-ulm 25720 df-log 25896 df-cxp 25897 |
This theorem is referenced by: (None) |
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