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Theorem stirlinglem2 37510
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 10865 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
2 faccl 12455 . . . . 5  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
3 nnrp 11300 . . . . 5  |-  ( ( ! `  N )  e.  NN  ->  ( ! `  N )  e.  RR+ )
41, 2, 33syl 18 . . . 4  |-  ( N  e.  NN  ->  ( ! `  N )  e.  RR+ )
5 2rp 11296 . . . . . . . 8  |-  2  e.  RR+
65a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  2  e.  RR+ )
7 nnrp 11300 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR+ )
86, 7rpmulcld 11346 . . . . . 6  |-  ( N  e.  NN  ->  (
2  x.  N )  e.  RR+ )
98rpsqrtcld 13441 . . . . 5  |-  ( N  e.  NN  ->  ( sqr `  ( 2  x.  N ) )  e.  RR+ )
10 epr 14227 . . . . . . . 8  |-  _e  e.  RR+
1110a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  _e  e.  RR+ )
127, 11rpdivcld 11347 . . . . . 6  |-  ( N  e.  NN  ->  ( N  /  _e )  e.  RR+ )
13 nnz 10948 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
1412, 13rpexpcld 12425 . . . . 5  |-  ( N  e.  NN  ->  (
( N  /  _e ) ^ N )  e.  RR+ )
159, 14rpmulcld 11346 . . . 4  |-  ( N  e.  NN  ->  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) )  e.  RR+ )
164, 15rpdivcld 11347 . . 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 5872 . . . . . . . 8  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
19 oveq2 6304 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2  x.  n )  =  ( 2  x.  k ) )
2019fveq2d 5876 . . . . . . . . 9  |-  ( n  =  k  ->  ( sqr `  ( 2  x.  n ) )  =  ( sqr `  (
2  x.  k ) ) )
21 oveq1 6303 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  /  _e )  =  ( k  /  _e ) )
22 id 23 . . . . . . . . . 10  |-  ( n  =  k  ->  n  =  k )
2321, 22oveq12d 6314 . . . . . . . . 9  |-  ( n  =  k  ->  (
( n  /  _e ) ^ n )  =  ( ( k  /  _e ) ^ k ) )
2420, 23oveq12d 6314 . . . . . . . 8  |-  ( n  =  k  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )
2518, 24oveq12d 6314 . . . . . . 7  |-  ( n  =  k  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  =  ( ( ! `
 k )  / 
( ( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
2625cbvmptv 4509 . . . . . 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 2449 . . . . 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 462 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  k  =  N )
3029fveq2d 5876 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( ! `  k )  =  ( ! `  N ) )
3129oveq2d 6312 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( 2  x.  k )  =  ( 2  x.  N
) )
3231fveq2d 5876 . . . . . 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 6311 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( k  /  _e )  =  ( N  /  _e ) )
3433, 29oveq12d 6314 . . . . . 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 6314 . . . . 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 6314 . . . 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 458 . . . 4  |-  ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  ->  N  e.  NN )
38 simpr 462 . . . 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 5961 . . 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 672 . 2  |-  ( N  e.  NN  ->  ( A `  N )  =  ( ( ! `
 N )  / 
( ( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) ) )
4140, 16eqeltrd 2508 1  |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867    |-> cmpt 4475   ` cfv 5592  (class class class)co 6296    x. cmul 9533    / cdiv 10258   NNcn 10598   2c2 10648   NN0cn0 10858   RR+crp 11291   ^cexp 12258   !cfa 12445   sqrcsqrt 13264   _eceu 14082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-n0 10859  df-z 10927  df-uz 11149  df-q 11254  df-rp 11292  df-ico 11630  df-fz 11772  df-fzo 11903  df-fl 12014  df-seq 12200  df-exp 12259  df-fac 12446  df-bc 12474  df-hash 12502  df-shft 13098  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-limsup 13493  df-clim 13519  df-rlim 13520  df-sum 13720  df-ef 14088  df-e 14089
This theorem is referenced by:  stirlinglem4  37512  stirlinglem11  37519  stirlinglem12  37520  stirlinglem13  37521  stirlinglem14  37522
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