Theorem iprodfac 31633

Description: An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)

Assertion

Ref	Expression
iprodfac	|- ( A e. NN0 -> ( ! ` A ) = prod_ k e. NN ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) )

Distinct variable group:   A, k

Proof of Theorem iprodfac

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 11408	. . 3 |- ( A e. NN0 -> 1 e. ZZ )
3		facne0 13073	. . 3 |- ( A e. NN0 -> ( ! ` A ) =/= 0 )
4		eqid 2622	. . . 4 |- ( x e. NN |-> ( ( ( 1 + ( 1 / x ) ) ^ A ) / ( 1 + ( A / x ) ) ) ) = ( x e. NN |-> ( ( ( 1 + ( 1 / x ) ) ^ A ) / ( 1 + ( A / x ) ) ) )
5	4	faclim 31632	. . 3 |- ( A e. NN0 -> seq 1 ( x. , ( x e. NN |-> ( ( ( 1 + ( 1 / x ) ) ^ A ) / ( 1 + ( A / x ) ) ) ) ) ~~> ( ! ` A ) )
6		oveq2 6658	. . . . . . . 8 |- ( x = k -> ( 1 / x ) = ( 1 / k ) )
7	6	oveq2d 6666	. . . . . . 7 |- ( x = k -> ( 1 + ( 1 / x ) ) = ( 1 + ( 1 / k ) ) )
8	7	oveq1d 6665	. . . . . 6 |- ( x = k -> ( ( 1 + ( 1 / x ) ) ^ A ) = ( ( 1 + ( 1 / k ) ) ^ A ) )
9		oveq2 6658	. . . . . . 7 |- ( x = k -> ( A / x ) = ( A / k ) )
10	9	oveq2d 6666	. . . . . 6 |- ( x = k -> ( 1 + ( A / x ) ) = ( 1 + ( A / k ) ) )
11	8, 10	oveq12d 6668	. . . . 5 |- ( x = k -> ( ( ( 1 + ( 1 / x ) ) ^ A ) / ( 1 + ( A / x ) ) ) = ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) )
12		ovex 6678	. . . . 5 |- ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) e. _V
13	11, 4, 12	fvmpt 6282	. . . 4 |- ( k e. NN -> ( ( x e. NN |-> ( ( ( 1 + ( 1 / x ) ) ^ A ) / ( 1 + ( A / x ) ) ) ) ` k ) = ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) )
14	13	adantl 482	. . 3 |- ( ( A e. NN0 /\ k e. NN ) -> ( ( x e. NN |-> ( ( ( 1 + ( 1 / x ) ) ^ A ) / ( 1 + ( A / x ) ) ) ) ` k ) = ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) )
15		1rp 11836	. . . . . . . 8 |- 1 e. RR+
16	15	a1i 11	. . . . . . 7 |- ( ( A e. NN0 /\ k e. NN ) -> 1 e. RR+ )
17		simpr 477	. . . . . . . . 9 |- ( ( A e. NN0 /\ k e. NN ) -> k e. NN )
18	17	nnrpd 11870	. . . . . . . 8 |- ( ( A e. NN0 /\ k e. NN ) -> k e. RR+ )
19	18	rpreccld 11882	. . . . . . 7 |- ( ( A e. NN0 /\ k e. NN ) -> ( 1 / k ) e. RR+ )
20	16, 19	rpaddcld 11887	. . . . . 6 |- ( ( A e. NN0 /\ k e. NN ) -> ( 1 + ( 1 / k ) ) e. RR+ )
21		nn0z 11400	. . . . . . 7 |- ( A e. NN0 -> A e. ZZ )
22	21	adantr 481	. . . . . 6 |- ( ( A e. NN0 /\ k e. NN ) -> A e. ZZ )
23	20, 22	rpexpcld 13032	. . . . 5 |- ( ( A e. NN0 /\ k e. NN ) -> ( ( 1 + ( 1 / k ) ) ^ A ) e. RR+ )
24		1cnd 10056	. . . . . . 7 |- ( ( A e. NN0 /\ k e. NN ) -> 1 e. CC )
25		nn0nndivcl 11362	. . . . . . . 8 |- ( ( A e. NN0 /\ k e. NN ) -> ( A / k ) e. RR )
26	25	recnd 10068	. . . . . . 7 |- ( ( A e. NN0 /\ k e. NN ) -> ( A / k ) e. CC )
27	24, 26	addcomd 10238	. . . . . 6 |- ( ( A e. NN0 /\ k e. NN ) -> ( 1 + ( A / k ) ) = ( ( A / k ) + 1 ) )
28		nn0ge0div 11446	. . . . . . 7 |- ( ( A e. NN0 /\ k e. NN ) -> 0 <_ ( A / k ) )
29	25, 28	ge0p1rpd 11902	. . . . . 6 |- ( ( A e. NN0 /\ k e. NN ) -> ( ( A / k ) + 1 ) e. RR+ )
30	27, 29	eqeltrd 2701	. . . . 5 |- ( ( A e. NN0 /\ k e. NN ) -> ( 1 + ( A / k ) ) e. RR+ )
31	23, 30	rpdivcld 11889	. . . 4 |- ( ( A e. NN0 /\ k e. NN ) -> ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) e. RR+ )
32	31	rpcnd 11874	. . 3 |- ( ( A e. NN0 /\ k e. NN ) -> ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) e. CC )
33	1, 2, 3, 5, 14, 32	iprodn0 14670	. 2 |- ( A e. NN0 -> prod_ k e. NN ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) = ( ! ` A ) )
34	33	eqcomd 2628	1 |- ( A e. NN0 -> ( ! ` A ) = prod_ k e. NN ( ( ( 1 + ( 1 / k ) ) ^ A ) / ( 1 + ( A / k ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832   ^cexp 12860   !cfa 13060   prod_cprod 14635

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-prod 14636

This theorem is referenced by: (None)