Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stirlinglem2 Structured version   Unicode version

Theorem stirlinglem2 29882
Description:  A maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Hypothesis
Ref Expression
stirlinglem2.1  |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
Assertion
Ref Expression
stirlinglem2  |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )

Proof of Theorem stirlinglem2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nnnn0 10598 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
2 faccl 12073 . . . . 5  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
3 nnrp 11012 . . . . 5  |-  ( ( ! `  N )  e.  NN  ->  ( ! `  N )  e.  RR+ )
41, 2, 33syl 20 . . . 4  |-  ( N  e.  NN  ->  ( ! `  N )  e.  RR+ )
5 2rp 11008 . . . . . . . 8  |-  2  e.  RR+
65a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  2  e.  RR+ )
7 nnrp 11012 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR+ )
86, 7rpmulcld 11055 . . . . . 6  |-  ( N  e.  NN  ->  (
2  x.  N )  e.  RR+ )
98rpsqrcld 12910 . . . . 5  |-  ( N  e.  NN  ->  ( sqr `  ( 2  x.  N ) )  e.  RR+ )
10 epr 13502 . . . . . . . 8  |-  _e  e.  RR+
1110a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  _e  e.  RR+ )
127, 11rpdivcld 11056 . . . . . 6  |-  ( N  e.  NN  ->  ( N  /  _e )  e.  RR+ )
13 nnz 10680 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
1412, 13rpexpcld 12043 . . . . 5  |-  ( N  e.  NN  ->  (
( N  /  _e ) ^ N )  e.  RR+ )
159, 14rpmulcld 11055 . . . 4  |-  ( N  e.  NN  ->  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) )  e.  RR+ )
164, 15rpdivcld 11056 . . 3  |-  ( N  e.  NN  ->  (
( ! `  N
)  /  ( ( sqr `  ( 2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )
17 stirlinglem2.1 . . . . . 6  |-  A  =  ( n  e.  NN  |->  ( ( ! `  n )  /  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) ) ) )
18 fveq2 5703 . . . . . . . 8  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
19 oveq2 6111 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2  x.  n )  =  ( 2  x.  k ) )
2019fveq2d 5707 . . . . . . . . 9  |-  ( n  =  k  ->  ( sqr `  ( 2  x.  n ) )  =  ( sqr `  (
2  x.  k ) ) )
21 oveq1 6110 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  /  _e )  =  ( k  /  _e ) )
22 id 22 . . . . . . . . . 10  |-  ( n  =  k  ->  n  =  k )
2321, 22oveq12d 6121 . . . . . . . . 9  |-  ( n  =  k  ->  (
( n  /  _e ) ^ n )  =  ( ( k  /  _e ) ^ k ) )
2420, 23oveq12d 6121 . . . . . . . 8  |-  ( n  =  k  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )
2518, 24oveq12d 6121 . . . . . . 7  |-  ( n  =  k  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  =  ( ( ! `
 k )  / 
( ( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
2625cbvmptv 4395 . . . . . 6  |-  ( n  e.  NN  |->  ( ( ! `  n )  /  ( ( sqr `  ( 2  x.  n
) )  x.  (
( n  /  _e ) ^ n ) ) ) )  =  ( k  e.  NN  |->  ( ( ! `  k
)  /  ( ( sqr `  ( 2  x.  k ) )  x.  ( ( k  /  _e ) ^
k ) ) ) )
2717, 26eqtri 2463 . . . . 5  |-  A  =  ( k  e.  NN  |->  ( ( ! `  k )  /  (
( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
2827a1i 11 . . . 4  |-  ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  ->  A  =  ( k  e.  NN  |->  ( ( ! `  k )  /  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) ) ) )
29 simpr 461 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  k  =  N )
3029fveq2d 5707 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( ! `  k )  =  ( ! `  N ) )
3129oveq2d 6119 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( 2  x.  k )  =  ( 2  x.  N
) )
3231fveq2d 5707 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( sqr `  ( 2  x.  k
) )  =  ( sqr `  ( 2  x.  N ) ) )
3329oveq1d 6118 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( k  /  _e )  =  ( N  /  _e ) )
3433, 29oveq12d 6121 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( (
k  /  _e ) ^ k )  =  ( ( N  /  _e ) ^ N ) )
3532, 34oveq12d 6121 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( ( sqr `  ( 2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) )  =  ( ( sqr `  ( 2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )
3630, 35oveq12d 6121 . . . 4  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( ( ! `  k )  /  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )  =  ( ( ! `  N )  /  ( ( sqr `  ( 2  x.  N
) )  x.  (
( N  /  _e ) ^ N ) ) ) )
37 simpl 457 . . . 4  |-  ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  ->  N  e.  NN )
38 simpr 461 . . . 4  |-  ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  ->  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )
3928, 36, 37, 38fvmptd 5791 . . 3  |-  ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  ->  ( A `  N
)  =  ( ( ! `  N )  /  ( ( sqr `  ( 2  x.  N
) )  x.  (
( N  /  _e ) ^ N ) ) ) )
4016, 39mpdan 668 . 2  |-  ( N  e.  NN  ->  ( A `  N )  =  ( ( ! `
 N )  / 
( ( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) ) )
4140, 16eqeltrd 2517 1  |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4362   ` cfv 5430  (class class class)co 6103    x. cmul 9299    / cdiv 10005   NNcn 10334   2c2 10383   NN0cn0 10591   RR+crp 11003   ^cexp 11877   !cfa 12063   sqrcsqr 12734   _eceu 13360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-ico 11318  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-shft 12568  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-limsup 12961  df-clim 12978  df-rlim 12979  df-sum 13176  df-ef 13365  df-e 13366
This theorem is referenced by:  stirlinglem4  29884  stirlinglem11  29891  stirlinglem12  29892  stirlinglem13  29893
  Copyright terms: Public domain W3C validator