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Mirrors > Home > MPE Home > Th. List > gamcvg | Structured version Visualization version GIF version |
Description: The pointwise exponential of the series πΊ converges to Ξ(π΄) Β· π΄. (Contributed by Mario Carneiro, 6-Jul-2017.) |
Ref | Expression |
---|---|
lgamcvg.g | β’ πΊ = (π β β β¦ ((π΄ Β· (logβ((π + 1) / π))) β (logβ((π΄ / π) + 1)))) |
lgamcvg.a | β’ (π β π΄ β (β β (β€ β β))) |
Ref | Expression |
---|---|
gamcvg | β’ (π β (exp β seq1( + , πΊ)) β ((Ξβπ΄) Β· π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 12866 | . . 3 β’ β = (β€β₯β1) | |
2 | 1zzd 12594 | . . 3 β’ (π β 1 β β€) | |
3 | efcn 26331 | . . . 4 β’ exp β (ββcnββ) | |
4 | 3 | a1i 11 | . . 3 β’ (π β exp β (ββcnββ)) |
5 | lgamcvg.a | . . . . . . . . . 10 β’ (π β π΄ β (β β (β€ β β))) | |
6 | 5 | eldifad 3955 | . . . . . . . . 9 β’ (π β π΄ β β) |
7 | 6 | adantr 480 | . . . . . . . 8 β’ ((π β§ π β β) β π΄ β β) |
8 | simpr 484 | . . . . . . . . . . . . 13 β’ ((π β§ π β β) β π β β) | |
9 | 8 | peano2nnd 12230 | . . . . . . . . . . . 12 β’ ((π β§ π β β) β (π + 1) β β) |
10 | 9 | nnrpd 13017 | . . . . . . . . . . 11 β’ ((π β§ π β β) β (π + 1) β β+) |
11 | 8 | nnrpd 13017 | . . . . . . . . . . 11 β’ ((π β§ π β β) β π β β+) |
12 | 10, 11 | rpdivcld 13036 | . . . . . . . . . 10 β’ ((π β§ π β β) β ((π + 1) / π) β β+) |
13 | 12 | relogcld 26508 | . . . . . . . . 9 β’ ((π β§ π β β) β (logβ((π + 1) / π)) β β) |
14 | 13 | recnd 11243 | . . . . . . . 8 β’ ((π β§ π β β) β (logβ((π + 1) / π)) β β) |
15 | 7, 14 | mulcld 11235 | . . . . . . 7 β’ ((π β§ π β β) β (π΄ Β· (logβ((π + 1) / π))) β β) |
16 | 8 | nncnd 12229 | . . . . . . . . . 10 β’ ((π β§ π β β) β π β β) |
17 | 8 | nnne0d 12263 | . . . . . . . . . 10 β’ ((π β§ π β β) β π β 0) |
18 | 7, 16, 17 | divcld 11991 | . . . . . . . . 9 β’ ((π β§ π β β) β (π΄ / π) β β) |
19 | 1cnd 11210 | . . . . . . . . 9 β’ ((π β§ π β β) β 1 β β) | |
20 | 18, 19 | addcld 11234 | . . . . . . . 8 β’ ((π β§ π β β) β ((π΄ / π) + 1) β β) |
21 | 5 | adantr 480 | . . . . . . . . 9 β’ ((π β§ π β β) β π΄ β (β β (β€ β β))) |
22 | 21, 8 | dmgmdivn0 26911 | . . . . . . . 8 β’ ((π β§ π β β) β ((π΄ / π) + 1) β 0) |
23 | 20, 22 | logcld 26455 | . . . . . . 7 β’ ((π β§ π β β) β (logβ((π΄ / π) + 1)) β β) |
24 | 15, 23 | subcld 11572 | . . . . . 6 β’ ((π β§ π β β) β ((π΄ Β· (logβ((π + 1) / π))) β (logβ((π΄ / π) + 1))) β β) |
25 | lgamcvg.g | . . . . . 6 β’ πΊ = (π β β β¦ ((π΄ Β· (logβ((π + 1) / π))) β (logβ((π΄ / π) + 1)))) | |
26 | 24, 25 | fmptd 7108 | . . . . 5 β’ (π β πΊ:ββΆβ) |
27 | 26 | ffvelcdmda 7079 | . . . 4 β’ ((π β§ π β β) β (πΊβπ) β β) |
28 | 1, 2, 27 | serf 13999 | . . 3 β’ (π β seq1( + , πΊ):ββΆβ) |
29 | 25, 5 | lgamcvg 26937 | . . 3 β’ (π β seq1( + , πΊ) β ((log Ξβπ΄) + (logβπ΄))) |
30 | lgamcl 26924 | . . . . 5 β’ (π΄ β (β β (β€ β β)) β (log Ξβπ΄) β β) | |
31 | 5, 30 | syl 17 | . . . 4 β’ (π β (log Ξβπ΄) β β) |
32 | 5 | dmgmn0 26909 | . . . . 5 β’ (π β π΄ β 0) |
33 | 6, 32 | logcld 26455 | . . . 4 β’ (π β (logβπ΄) β β) |
34 | 31, 33 | addcld 11234 | . . 3 β’ (π β ((log Ξβπ΄) + (logβπ΄)) β β) |
35 | 1, 2, 4, 28, 29, 34 | climcncf 24771 | . 2 β’ (π β (exp β seq1( + , πΊ)) β (expβ((log Ξβπ΄) + (logβπ΄)))) |
36 | efadd 16042 | . . . 4 β’ (((log Ξβπ΄) β β β§ (logβπ΄) β β) β (expβ((log Ξβπ΄) + (logβπ΄))) = ((expβ(log Ξβπ΄)) Β· (expβ(logβπ΄)))) | |
37 | 31, 33, 36 | syl2anc 583 | . . 3 β’ (π β (expβ((log Ξβπ΄) + (logβπ΄))) = ((expβ(log Ξβπ΄)) Β· (expβ(logβπ΄)))) |
38 | eflgam 26928 | . . . . 5 β’ (π΄ β (β β (β€ β β)) β (expβ(log Ξβπ΄)) = (Ξβπ΄)) | |
39 | 5, 38 | syl 17 | . . . 4 β’ (π β (expβ(log Ξβπ΄)) = (Ξβπ΄)) |
40 | eflog 26461 | . . . . 5 β’ ((π΄ β β β§ π΄ β 0) β (expβ(logβπ΄)) = π΄) | |
41 | 6, 32, 40 | syl2anc 583 | . . . 4 β’ (π β (expβ(logβπ΄)) = π΄) |
42 | 39, 41 | oveq12d 7422 | . . 3 β’ (π β ((expβ(log Ξβπ΄)) Β· (expβ(logβπ΄))) = ((Ξβπ΄) Β· π΄)) |
43 | 37, 42 | eqtrd 2766 | . 2 β’ (π β (expβ((log Ξβπ΄) + (logβπ΄))) = ((Ξβπ΄) Β· π΄)) |
44 | 35, 43 | breqtrd 5167 | 1 β’ (π β (exp β seq1( + , πΊ)) β ((Ξβπ΄) Β· π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 β cdif 3940 class class class wbr 5141 β¦ cmpt 5224 β ccom 5673 βcfv 6536 (class class class)co 7404 βcc 11107 0cc0 11109 1c1 11110 + caddc 11112 Β· cmul 11114 β cmin 11445 / cdiv 11872 βcn 12213 β€cz 12559 seqcseq 13969 β cli 15432 expce 16009 βcnβccncf 24747 logclog 26439 log Ξclgam 26899 Ξcgam 26900 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-tan 16019 df-pi 16020 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-cmp 23242 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-limc 25746 df-dv 25747 df-ulm 26264 df-log 26441 df-cxp 26442 df-lgam 26902 df-gam 26903 |
This theorem is referenced by: gamcvg2 26943 |
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