Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fallfacval4 Structured version   Unicode version

Theorem fallfacval4 28742
Description: Represent the falling factorial via factorials when the first argument is a natural. (Contributed by Scott Fenton, 20-Mar-2018.)
Assertion
Ref Expression
fallfacval4  |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  ( ( ! `  A
)  /  ( ! `
 ( A  -  N ) ) ) )

Proof of Theorem fallfacval4
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fzfid 12047 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
( ( A  -  N )  +  1 ) ... A )  e.  Fin )
2 elfzelz 11684 . . . . . . 7  |-  ( k  e.  ( ( ( A  -  N )  +  1 ) ... A )  ->  k  e.  ZZ )
32zcnd 10963 . . . . . 6  |-  ( k  e.  ( ( ( A  -  N )  +  1 ) ... A )  ->  k  e.  CC )
43adantl 466 . . . . 5  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( (
( A  -  N
)  +  1 ) ... A ) )  ->  k  e.  CC )
51, 4fprodcl 28661 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( ( ( A  -  N )  +  1 ) ... A ) k  e.  CC )
6 fzfid 12047 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
1 ... ( A  -  N ) )  e. 
Fin )
7 elfznn 11710 . . . . . . 7  |-  ( k  e.  ( 1 ... ( A  -  N
) )  ->  k  e.  NN )
87adantl 466 . . . . . 6  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( 1 ... ( A  -  N ) ) )  ->  k  e.  NN )
98nncnd 10548 . . . . 5  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( 1 ... ( A  -  N ) ) )  ->  k  e.  CC )
106, 9fprodcl 28661 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( 1 ... ( A  -  N
) ) k  e.  CC )
118nnne0d 10576 . . . . 5  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( 1 ... ( A  -  N ) ) )  ->  k  =/=  0
)
126, 9, 11fprodn0 28686 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( 1 ... ( A  -  N
) ) k  =/=  0 )
135, 10, 12divcan3d 10321 . . 3  |-  ( N  e.  ( 0 ... A )  ->  (
( prod_ k  e.  ( 1 ... ( A  -  N ) ) k  x.  prod_ k  e.  ( ( ( A  -  N )  +  1 ) ... A
) k )  /  prod_ k  e.  ( 1 ... ( A  -  N ) ) k )  =  prod_ k  e.  ( ( ( A  -  N )  +  1 ) ... A
) k )
14 fznn0sub 11712 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  NN0 )
1514nn0red 10849 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  RR )
1615ltp1d 10472 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  <  ( ( A  -  N )  +  1 ) )
17 fzdisj 11708 . . . . . 6  |-  ( ( A  -  N )  <  ( ( A  -  N )  +  1 )  ->  (
( 1 ... ( A  -  N )
)  i^i  ( (
( A  -  N
)  +  1 ) ... A ) )  =  (/) )
1816, 17syl 16 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
( 1 ... ( A  -  N )
)  i^i  ( (
( A  -  N
)  +  1 ) ... A ) )  =  (/) )
19 nn0p1nn 10831 . . . . . . . 8  |-  ( ( A  -  N )  e.  NN0  ->  ( ( A  -  N )  +  1 )  e.  NN )
2014, 19syl 16 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  (
( A  -  N
)  +  1 )  e.  NN )
21 nnuz 11113 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2220, 21syl6eleq 2565 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  (
( A  -  N
)  +  1 )  e.  ( ZZ>= `  1
) )
2314nn0zd 10960 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  ZZ )
24 elfzel2 11682 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  A  e.  ZZ )
25 elfzle1 11685 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  0  <_  N )
2624zred 10962 . . . . . . . . 9  |-  ( N  e.  ( 0 ... A )  ->  A  e.  RR )
27 elfzelz 11684 . . . . . . . . . 10  |-  ( N  e.  ( 0 ... A )  ->  N  e.  ZZ )
2827zred 10962 . . . . . . . . 9  |-  ( N  e.  ( 0 ... A )  ->  N  e.  RR )
2926, 28subge02d 10140 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  (
0  <_  N  <->  ( A  -  N )  <_  A
) )
3025, 29mpbid 210 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  <_  A )
31 eluz2 11084 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  ( A  -  N )
)  <->  ( ( A  -  N )  e.  ZZ  /\  A  e.  ZZ  /\  ( A  -  N )  <_  A ) )
3223, 24, 30, 31syl3anbrc 1180 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  A  e.  ( ZZ>= `  ( A  -  N ) ) )
33 fzsplit2 11706 . . . . . 6  |-  ( ( ( ( A  -  N )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  A  e.  ( ZZ>= `  ( A  -  N ) ) )  ->  ( 1 ... A )  =  ( ( 1 ... ( A  -  N )
)  u.  ( ( ( A  -  N
)  +  1 ) ... A ) ) )
3422, 32, 33syl2anc 661 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
1 ... A )  =  ( ( 1 ... ( A  -  N
) )  u.  (
( ( A  -  N )  +  1 ) ... A ) ) )
35 fzfid 12047 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
1 ... A )  e. 
Fin )
36 elfznn 11710 . . . . . . 7  |-  ( k  e.  ( 1 ... A )  ->  k  e.  NN )
3736nncnd 10548 . . . . . 6  |-  ( k  e.  ( 1 ... A )  ->  k  e.  CC )
3837adantl 466 . . . . 5  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( 1 ... A ) )  ->  k  e.  CC )
3918, 34, 35, 38fprodsplit 28672 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( 1 ... A ) k  =  ( prod_ k  e.  ( 1 ... ( A  -  N ) ) k  x.  prod_ k  e.  ( ( ( A  -  N )  +  1 ) ... A
) k ) )
4039oveq1d 6297 . . 3  |-  ( N  e.  ( 0 ... A )  ->  ( prod_ k  e.  ( 1 ... A ) k  /  prod_ k  e.  ( 1 ... ( A  -  N ) ) k )  =  ( ( prod_ k  e.  ( 1 ... ( A  -  N ) ) k  x.  prod_ k  e.  ( ( ( A  -  N )  +  1 ) ... A
) k )  /  prod_ k  e.  ( 1 ... ( A  -  N ) ) k ) )
4124zcnd 10963 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  A  e.  CC )
4227zcnd 10963 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  N  e.  CC )
43 ax-1cn 9546 . . . . . . 7  |-  1  e.  CC
4443a1i 11 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  1  e.  CC )
4541, 42, 44subsubd 9954 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  ( N  -  1 ) )  =  ( ( A  -  N )  +  1 ) )
4645oveq1d 6297 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  (
( A  -  ( N  -  1 ) ) ... A )  =  ( ( ( A  -  N )  +  1 ) ... A ) )
4746prodeq1d 28630 . . 3  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( ( A  -  ( N  - 
1 ) ) ... A ) k  = 
prod_ k  e.  (
( ( A  -  N )  +  1 ) ... A ) k )
4813, 40, 473eqtr4rd 2519 . 2  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( ( A  -  ( N  - 
1 ) ) ... A ) k  =  ( prod_ k  e.  ( 1 ... A ) k  /  prod_ k  e.  ( 1 ... ( A  -  N )
) k ) )
49 fallfacval3 28711 . 2  |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  prod_ k  e.  ( ( A  -  ( N  - 
1 ) ) ... A ) k )
50 elfz3nn0 11767 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  A  e.  NN0 )
51 fprodfac 28679 . . . 4  |-  ( A  e.  NN0  ->  ( ! `
 A )  = 
prod_ k  e.  (
1 ... A ) k )
5250, 51syl 16 . . 3  |-  ( N  e.  ( 0 ... A )  ->  ( ! `  A )  =  prod_ k  e.  ( 1 ... A ) k )
53 fprodfac 28679 . . . 4  |-  ( ( A  -  N )  e.  NN0  ->  ( ! `
 ( A  -  N ) )  = 
prod_ k  e.  (
1 ... ( A  -  N ) ) k )
5414, 53syl 16 . . 3  |-  ( N  e.  ( 0 ... A )  ->  ( ! `  ( A  -  N ) )  = 
prod_ k  e.  (
1 ... ( A  -  N ) ) k )
5552, 54oveq12d 6300 . 2  |-  ( N  e.  ( 0 ... A )  ->  (
( ! `  A
)  /  ( ! `
 ( A  -  N ) ) )  =  ( prod_ k  e.  ( 1 ... A
) k  /  prod_ k  e.  ( 1 ... ( A  -  N
) ) k ) )
5648, 49, 553eqtr4d 2518 1  |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  ( ( ! `  A
)  /  ( ! `
 ( A  -  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    i^i cin 3475   (/)c0 3785   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    <_ cle 9625    - cmin 9801    / cdiv 10202   NNcn 10532   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668   !cfa 12317   prod_cprod 28614   FallFac cfallfac 28703
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 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-fac 12318  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-prod 28615  df-fallfac 28706
This theorem is referenced by:  bcfallfac  28743  fallfacfac  28744
  Copyright terms: Public domain W3C validator