Theorem stirlinglem2 40292

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.

Step	Hyp	Ref	Expression
1		nnnn0 11299	. . . . 5 |- ( N e. NN -> N e. NN0 )
2		faccl 13070	. . . . 5 |- ( N e. NN0 -> ( ! ` N ) e. NN )
3		nnrp 11842	. . . . 5 |- ( ( ! ` N ) e. NN -> ( ! ` N ) e. RR+ )
4	1, 2, 3	3syl 18	. . . 4 |- ( N e. NN -> ( ! ` N ) e. RR+ )
5		2rp 11837	. . . . . . . 8 |- 2 e. RR+
6	5	a1i 11	. . . . . . 7 |- ( N e. NN -> 2 e. RR+ )
7		nnrp 11842	. . . . . . 7 |- ( N e. NN -> N e. RR+ )
8	6, 7	rpmulcld 11888	. . . . . 6 |- ( N e. NN -> ( 2 x. N ) e. RR+ )
9	8	rpsqrtcld 14150	. . . . 5 |- ( N e. NN -> ( sqr ` ( 2 x. N ) ) e. RR+ )
10		epr 14936	. . . . . . . 8 |- _e e. RR+
11	10	a1i 11	. . . . . . 7 |- ( N e. NN -> _e e. RR+ )
12	7, 11	rpdivcld 11889	. . . . . 6 |- ( N e. NN -> ( N / _e ) e. RR+ )
13		nnz 11399	. . . . . 6 |- ( N e. NN -> N e. ZZ )
14	12, 13	rpexpcld 13032	. . . . 5 |- ( N e. NN -> ( ( N / _e ) ^ N ) e. RR+ )
15	9, 14	rpmulcld 11888	. . . 4 |- ( N e. NN -> ( ( sqr ` ( 2 x. N ) ) x. ( ( N / _e ) ^ N ) ) e. RR+ )
16	4, 15	rpdivcld 11889	. . 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 6191	. . . . . . . 8 |- ( n = k -> ( ! ` n ) = ( ! ` k ) )
19		oveq2 6658	. . . . . . . . . 10 |- ( n = k -> ( 2 x. n ) = ( 2 x. k ) )
20	19	fveq2d 6195	. . . . . . . . 9 |- ( n = k -> ( sqr ` ( 2 x. n ) ) = ( sqr ` ( 2 x. k ) ) )
21		oveq1 6657	. . . . . . . . . 10 |- ( n = k -> ( n / _e ) = ( k / _e ) )
22		id 22	. . . . . . . . . 10 |- ( n = k -> n = k )
23	21, 22	oveq12d 6668	. . . . . . . . 9 |- ( n = k -> ( ( n / _e ) ^ n ) = ( ( k / _e ) ^ k ) )
24	20, 23	oveq12d 6668	. . . . . . . 8 |- ( n = k -> ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) = ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) )
25	18, 24	oveq12d 6668	. . . . . . 7 |- ( n = k -> ( ( ! ` n ) / ( ( sqr ` ( 2 x. n ) ) x. ( ( n / _e ) ^ n ) ) ) = ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) )
26	25	cbvmptv 4750	. . . . . 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 ) ) ) )
27	17, 26	eqtri 2644	. . . . 5 |- A = ( k e. NN |-> ( ( ! ` k ) / ( ( sqr ` ( 2 x. k ) ) x. ( ( k / _e ) ^ k ) ) ) )
28	27	a1i 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 477	. . . . . 6 |- ( ( ( N e. NN /\ ( ( ! ` N ) / ( ( sqr ` ( 2 x. N ) ) x. ( ( N / _e ) ^ N ) ) ) e. RR+ ) /\ k = N ) -> k = N )
30	29	fveq2d 6195	. . . . 5 |- ( ( ( N e. NN /\ ( ( ! ` N ) / ( ( sqr ` ( 2 x. N ) ) x. ( ( N / _e ) ^ N ) ) ) e. RR+ ) /\ k = N ) -> ( ! ` k ) = ( ! ` N ) )
31	29	oveq2d 6666	. . . . . . 7 |- ( ( ( N e. NN /\ ( ( ! ` N ) / ( ( sqr ` ( 2 x. N ) ) x. ( ( N / _e ) ^ N ) ) ) e. RR+ ) /\ k = N ) -> ( 2 x. k ) = ( 2 x. N ) )
32	31	fveq2d 6195	. . . . . 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 ) ) )
33	29	oveq1d 6665	. . . . . . 7 |- ( ( ( N e. NN /\ ( ( ! ` N ) / ( ( sqr ` ( 2 x. N ) ) x. ( ( N / _e ) ^ N ) ) ) e. RR+ ) /\ k = N ) -> ( k / _e ) = ( N / _e ) )
34	33, 29	oveq12d 6668	. . . . . 6 |- ( ( ( N e. NN /\ ( ( ! ` N ) / ( ( sqr ` ( 2 x. N ) ) x. ( ( N / _e ) ^ N ) ) ) e. RR+ ) /\ k = N ) -> ( ( k / _e ) ^ k ) = ( ( N / _e ) ^ N ) )
35	32, 34	oveq12d 6668	. . . . 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 ) ) )
36	30, 35	oveq12d 6668	. . . 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 473	. . . 4 |- ( ( N e. NN /\ ( ( ! ` N ) / ( ( sqr ` ( 2 x. N ) ) x. ( ( N / _e ) ^ N ) ) ) e. RR+ ) -> N e. NN )
38		simpr 477	. . . 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+ )
39	28, 36, 37, 38	fvmptd 6288	. . 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 ) ) ) )
40	16, 39	mpdan 702	. 2 |- ( N e. NN -> ( A ` N ) = ( ( ! ` N ) / ( ( sqr ` ( 2 x. N ) ) x. ( ( N / _e ) ^ N ) ) ) )
41	40, 16	eqeltrd 2701	1 |- ( N e. NN -> ( A ` N ) e. RR+ )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   x. cmul 9941   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   RR+crp 11832   ^cexp 12860   !cfa 13060   sqrcsqrt 13973   _eceu 14793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-e 14799

This theorem is referenced by:  stirlinglem4  40294  stirlinglem11  40301  stirlinglem12  40302  stirlinglem13  40303  stirlinglem14  40304