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Theorem stirlinglem2 31602
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 10803 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
2 faccl 12332 . . . . 5  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
3 nnrp 11230 . . . . 5  |-  ( ( ! `  N )  e.  NN  ->  ( ! `  N )  e.  RR+ )
41, 2, 33syl 20 . . . 4  |-  ( N  e.  NN  ->  ( ! `  N )  e.  RR+ )
5 2rp 11226 . . . . . . . 8  |-  2  e.  RR+
65a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  2  e.  RR+ )
7 nnrp 11230 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR+ )
86, 7rpmulcld 11273 . . . . . 6  |-  ( N  e.  NN  ->  (
2  x.  N )  e.  RR+ )
98rpsqrtcld 13209 . . . . 5  |-  ( N  e.  NN  ->  ( sqr `  ( 2  x.  N ) )  e.  RR+ )
10 epr 13805 . . . . . . . 8  |-  _e  e.  RR+
1110a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  _e  e.  RR+ )
127, 11rpdivcld 11274 . . . . . 6  |-  ( N  e.  NN  ->  ( N  /  _e )  e.  RR+ )
13 nnz 10887 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
1412, 13rpexpcld 12302 . . . . 5  |-  ( N  e.  NN  ->  (
( N  /  _e ) ^ N )  e.  RR+ )
159, 14rpmulcld 11273 . . . 4  |-  ( N  e.  NN  ->  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) )  e.  RR+ )
164, 15rpdivcld 11274 . . 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 5866 . . . . . . . 8  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
19 oveq2 6293 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2  x.  n )  =  ( 2  x.  k ) )
2019fveq2d 5870 . . . . . . . . 9  |-  ( n  =  k  ->  ( sqr `  ( 2  x.  n ) )  =  ( sqr `  (
2  x.  k ) ) )
21 oveq1 6292 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  /  _e )  =  ( k  /  _e ) )
22 id 22 . . . . . . . . . 10  |-  ( n  =  k  ->  n  =  k )
2321, 22oveq12d 6303 . . . . . . . . 9  |-  ( n  =  k  ->  (
( n  /  _e ) ^ n )  =  ( ( k  /  _e ) ^ k ) )
2420, 23oveq12d 6303 . . . . . . . 8  |-  ( n  =  k  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )
2518, 24oveq12d 6303 . . . . . . 7  |-  ( n  =  k  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  =  ( ( ! `
 k )  / 
( ( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
2625cbvmptv 4538 . . . . . 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 2496 . . . . 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 5870 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( ! `  k )  =  ( ! `  N ) )
3129oveq2d 6301 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( 2  x.  k )  =  ( 2  x.  N
) )
3231fveq2d 5870 . . . . . 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 6300 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( k  /  _e )  =  ( N  /  _e ) )
3433, 29oveq12d 6303 . . . . . 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 6303 . . . . 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 6303 . . . 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 5956 . . 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 2555 1  |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505   ` cfv 5588  (class class class)co 6285    x. cmul 9498    / cdiv 10207   NNcn 10537   2c2 10586   NN0cn0 10796   RR+crp 11221   ^cexp 12135   !cfa 12322   sqrcsqrt 13032   _eceu 13663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-n0 10797  df-z 10866  df-uz 11084  df-q 11184  df-rp 11222  df-ico 11536  df-fz 11674  df-fzo 11794  df-fl 11898  df-seq 12077  df-exp 12136  df-fac 12323  df-bc 12350  df-hash 12375  df-shft 12866  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-limsup 13260  df-clim 13277  df-rlim 13278  df-sum 13475  df-ef 13668  df-e 13669
This theorem is referenced by:  stirlinglem4  31604  stirlinglem11  31611  stirlinglem12  31612  stirlinglem13  31613
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