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Theorem stirlinglem2 37931
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 10873 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
2 faccl 12466 . . . . 5  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
3 nnrp 11308 . . . . 5  |-  ( ( ! `  N )  e.  NN  ->  ( ! `  N )  e.  RR+ )
41, 2, 33syl 18 . . . 4  |-  ( N  e.  NN  ->  ( ! `  N )  e.  RR+ )
5 2rp 11304 . . . . . . . 8  |-  2  e.  RR+
65a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  2  e.  RR+ )
7 nnrp 11308 . . . . . . 7  |-  ( N  e.  NN  ->  N  e.  RR+ )
86, 7rpmulcld 11354 . . . . . 6  |-  ( N  e.  NN  ->  (
2  x.  N )  e.  RR+ )
98rpsqrtcld 13466 . . . . 5  |-  ( N  e.  NN  ->  ( sqr `  ( 2  x.  N ) )  e.  RR+ )
10 epr 14253 . . . . . . . 8  |-  _e  e.  RR+
1110a1i 11 . . . . . . 7  |-  ( N  e.  NN  ->  _e  e.  RR+ )
127, 11rpdivcld 11355 . . . . . 6  |-  ( N  e.  NN  ->  ( N  /  _e )  e.  RR+ )
13 nnz 10956 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  ZZ )
1412, 13rpexpcld 12436 . . . . 5  |-  ( N  e.  NN  ->  (
( N  /  _e ) ^ N )  e.  RR+ )
159, 14rpmulcld 11354 . . . 4  |-  ( N  e.  NN  ->  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) )  e.  RR+ )
164, 15rpdivcld 11355 . . 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 5863 . . . . . . . 8  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
19 oveq2 6296 . . . . . . . . . 10  |-  ( n  =  k  ->  (
2  x.  n )  =  ( 2  x.  k ) )
2019fveq2d 5867 . . . . . . . . 9  |-  ( n  =  k  ->  ( sqr `  ( 2  x.  n ) )  =  ( sqr `  (
2  x.  k ) ) )
21 oveq1 6295 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  /  _e )  =  ( k  /  _e ) )
22 id 22 . . . . . . . . . 10  |-  ( n  =  k  ->  n  =  k )
2321, 22oveq12d 6306 . . . . . . . . 9  |-  ( n  =  k  ->  (
( n  /  _e ) ^ n )  =  ( ( k  /  _e ) ^ k ) )
2420, 23oveq12d 6306 . . . . . . . 8  |-  ( n  =  k  ->  (
( sqr `  (
2  x.  n ) )  x.  ( ( n  /  _e ) ^ n ) )  =  ( ( sqr `  ( 2  x.  k
) )  x.  (
( k  /  _e ) ^ k ) ) )
2518, 24oveq12d 6306 . . . . . . 7  |-  ( n  =  k  ->  (
( ! `  n
)  /  ( ( sqr `  ( 2  x.  n ) )  x.  ( ( n  /  _e ) ^
n ) ) )  =  ( ( ! `
 k )  / 
( ( sqr `  (
2  x.  k ) )  x.  ( ( k  /  _e ) ^ k ) ) ) )
2625cbvmptv 4494 . . . . . 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 2472 . . . . 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 463 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  k  =  N )
3029fveq2d 5867 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( ! `  k )  =  ( ! `  N ) )
3129oveq2d 6304 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( 2  x.  k )  =  ( 2  x.  N
) )
3231fveq2d 5867 . . . . . 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 6303 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  /\  k  =  N
)  ->  ( k  /  _e )  =  ( N  /  _e ) )
3433, 29oveq12d 6306 . . . . . 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 6306 . . . . 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 6306 . . . 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 459 . . . 4  |-  ( ( N  e.  NN  /\  ( ( ! `  N )  /  (
( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) )  e.  RR+ )  ->  N  e.  NN )
38 simpr 463 . . . 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 5952 . . 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 673 . 2  |-  ( N  e.  NN  ->  ( A `  N )  =  ( ( ! `
 N )  / 
( ( sqr `  (
2  x.  N ) )  x.  ( ( N  /  _e ) ^ N ) ) ) )
4140, 16eqeltrd 2528 1  |-  ( N  e.  NN  ->  ( A `  N )  e.  RR+ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886    |-> cmpt 4460   ` cfv 5581  (class class class)co 6288    x. cmul 9541    / cdiv 10266   NNcn 10606   2c2 10656   NN0cn0 10866   RR+crp 11299   ^cexp 12269   !cfa 12456   sqrcsqrt 13289   _eceu 14108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-ico 11638  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-e 14115
This theorem is referenced by:  stirlinglem4  37933  stirlinglem11  37940  stirlinglem12  37941  stirlinglem13  37942  stirlinglem14  37943
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