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Theorem stirlinglem2 27800
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 10228 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
2 faccl 11576 . . . . 5  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
3 nnrp 10621 . . . . 5  |-  ( ( ! `  N )  e.  NN  ->  ( ! `  N )  e.  RR+ )
41, 2, 33syl 19 . . . 4  |-  ( N  e.  NN  ->  ( ! `  N )  e.  RR+ )
5 2rp 10617 . . . . . . . 8  |-  2  e.  RR+
65a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  2  e.  RR+ )
7 nnrp 10621 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR+ )
86, 7rpmulcld 10664 . . . . . 6  |-  ( N  e.  NN  ->  (
2  x.  N )  e.  RR+ )
98rpsqrcld 12214 . . . . 5  |-  ( N  e.  NN  ->  ( sqr `  ( 2  x.  N ) )  e.  RR+ )
10 epr 12807 . . . . . . . 8  |-  _e  e.  RR+
1110a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  _e  e.  RR+ )
127, 11rpdivcld 10665 . . . . . 6  |-  ( N  e.  NN  ->  ( N  /  _e )  e.  RR+ )
13 nnz 10303 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
1412, 13rpexpcld 11546 . . . . 5  |-  ( N  e.  NN  ->  (
( N  /  _e ) ^ N )  e.  RR+ )
159, 14rpmulcld 10664 . . . 4  |-  ( N  e.  NN  ->  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) )  e.  RR+ )
164, 15rpdivcld 10665 . . 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 5728 . . . . . . . 8  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
19 oveq2 6089 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2  x.  n )  =  ( 2  x.  k ) )
2019fveq2d 5732 . . . . . . . . 9  |-  ( n  =  k  ->  ( sqr `  ( 2  x.  n ) )  =  ( sqr `  (
2  x.  k ) ) )
21 oveq1 6088 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  /  _e )  =  ( k  /  _e ) )
22 id 20 . . . . . . . . . 10  |-  ( n  =  k  ->  n  =  k )
2321, 22oveq12d 6099 . . . . . . . . 9  |-  ( n  =  k  ->  (
( n  /  _e ) ^ n )  =  ( ( k  /  _e ) ^ k ) )
2420, 23oveq12d 6099 . . . . . . . 8  |-  ( n  =  k  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )
2518, 24oveq12d 6099 . . . . . . 7  |-  ( n  =  k  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  =  ( ( ! `
 k )  / 
( ( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
2625cbvmptv 4300 . . . . . 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 2456 . . . . 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 448 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  k  =  N )
3029fveq2d 5732 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( ! `  k )  =  ( ! `  N ) )
3129oveq2d 6097 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( 2  x.  k )  =  ( 2  x.  N
) )
3231fveq2d 5732 . . . . . 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 6096 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( k  /  _e )  =  ( N  /  _e ) )
3433, 29oveq12d 6099 . . . . . 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 6099 . . . . 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 6099 . . . 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 444 . . . 4  |-  ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  ->  N  e.  NN )
38 simpr 448 . . . 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 5810 . . 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 650 . 2  |-  ( N  e.  NN  ->  ( A `  N )  =  ( ( ! `
 N )  / 
( ( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) ) )
4140, 16eqeltrd 2510 1  |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4266   ` cfv 5454  (class class class)co 6081    x. cmul 8995    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   RR+crp 10612   ^cexp 11382   !cfa 11566   sqrcsqr 12038   _eceu 12665
This theorem is referenced by:  stirlinglem4  27802  stirlinglem11  27809  stirlinglem12  27810  stirlinglem13  27811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-e 12671
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