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Theorem fallfacval4 13990
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 12126 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
( ( A  -  N )  +  1 ) ... A )  e.  Fin )
2 elfzelz 11744 . . . . . . 7  |-  ( k  e.  ( ( ( A  -  N )  +  1 ) ... A )  ->  k  e.  ZZ )
32zcnd 11011 . . . . . 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 13913 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( ( ( A  -  N )  +  1 ) ... A ) k  e.  CC )
6 fzfid 12126 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
1 ... ( A  -  N ) )  e. 
Fin )
7 elfznn 11770 . . . . . . 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 10594 . . . . 5  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( 1 ... ( A  -  N ) ) )  ->  k  e.  CC )
106, 9fprodcl 13913 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( 1 ... ( A  -  N
) ) k  e.  CC )
118nnne0d 10623 . . . . 5  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( 1 ... ( A  -  N ) ) )  ->  k  =/=  0
)
126, 9, 11fprodn0 13937 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( 1 ... ( A  -  N
) ) k  =/=  0 )
135, 10, 12divcan3d 10368 . . 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 11773 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  NN0 )
1514nn0red 10896 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  RR )
1615ltp1d 10518 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  <  ( ( A  -  N )  +  1 ) )
17 fzdisj 11768 . . . . . 6  |-  ( ( A  -  N )  <  ( ( A  -  N )  +  1 )  ->  (
( 1 ... ( A  -  N )
)  i^i  ( (
( A  -  N
)  +  1 ) ... A ) )  =  (/) )
1816, 17syl 17 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
( 1 ... ( A  -  N )
)  i^i  ( (
( A  -  N
)  +  1 ) ... A ) )  =  (/) )
19 nn0p1nn 10878 . . . . . . . 8  |-  ( ( A  -  N )  e.  NN0  ->  ( ( A  -  N )  +  1 )  e.  NN )
2014, 19syl 17 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  (
( A  -  N
)  +  1 )  e.  NN )
21 nnuz 11164 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2220, 21syl6eleq 2502 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  (
( A  -  N
)  +  1 )  e.  ( ZZ>= `  1
) )
2314nn0zd 11008 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  ZZ )
24 elfzel2 11742 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  A  e.  ZZ )
25 elfzle1 11745 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  0  <_  N )
2624zred 11010 . . . . . . . . 9  |-  ( N  e.  ( 0 ... A )  ->  A  e.  RR )
27 elfzelz 11744 . . . . . . . . . 10  |-  ( N  e.  ( 0 ... A )  ->  N  e.  ZZ )
2827zred 11010 . . . . . . . . 9  |-  ( N  e.  ( 0 ... A )  ->  N  e.  RR )
2926, 28subge02d 10186 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  (
0  <_  N  <->  ( A  -  N )  <_  A
) )
3025, 29mpbid 212 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  <_  A )
31 eluz2 11135 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  ( A  -  N )
)  <->  ( ( A  -  N )  e.  ZZ  /\  A  e.  ZZ  /\  ( A  -  N )  <_  A ) )
3223, 24, 30, 31syl3anbrc 1183 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  A  e.  ( ZZ>= `  ( A  -  N ) ) )
33 fzsplit2 11766 . . . . . 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 12126 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
1 ... A )  e. 
Fin )
36 elfznn 11770 . . . . . . 7  |-  ( k  e.  ( 1 ... A )  ->  k  e.  NN )
3736nncnd 10594 . . . . . 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 13924 . . . 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 6295 . . 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 11011 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  A  e.  CC )
4227zcnd 11011 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  N  e.  CC )
43 1cnd 9644 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  1  e.  CC )
4441, 42, 43subsubd 9997 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  ( N  -  1 ) )  =  ( ( A  -  N )  +  1 ) )
4544oveq1d 6295 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  (
( A  -  ( N  -  1 ) ) ... A )  =  ( ( ( A  -  N )  +  1 ) ... A ) )
4645prodeq1d 13882 . . 3  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( ( A  -  ( N  - 
1 ) ) ... A ) k  = 
prod_ k  e.  (
( ( A  -  N )  +  1 ) ... A ) k )
4713, 40, 463eqtr4rd 2456 . 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 ) )
48 fallfacval3 13959 . 2  |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  prod_ k  e.  ( ( A  -  ( N  - 
1 ) ) ... A ) k )
49 elfz3nn0 11829 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  A  e.  NN0 )
50 fprodfac 13931 . . . 4  |-  ( A  e.  NN0  ->  ( ! `
 A )  = 
prod_ k  e.  (
1 ... A ) k )
5149, 50syl 17 . . 3  |-  ( N  e.  ( 0 ... A )  ->  ( ! `  A )  =  prod_ k  e.  ( 1 ... A ) k )
52 fprodfac 13931 . . . 4  |-  ( ( A  -  N )  e.  NN0  ->  ( ! `
 ( A  -  N ) )  = 
prod_ k  e.  (
1 ... ( A  -  N ) ) k )
5314, 52syl 17 . . 3  |-  ( N  e.  ( 0 ... A )  ->  ( ! `  ( A  -  N ) )  = 
prod_ k  e.  (
1 ... ( A  -  N ) ) k )
5451, 53oveq12d 6298 . 2  |-  ( N  e.  ( 0 ... A )  ->  (
( ! `  A
)  /  ( ! `
 ( A  -  N ) ) )  =  ( prod_ k  e.  ( 1 ... A
) k  /  prod_ k  e.  ( 1 ... ( A  -  N
) ) k ) )
5547, 48, 543eqtr4d 2455 1  |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  ( ( ! `  A
)  /  ( ! `
 ( A  -  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    u. cun 3414    i^i cin 3415   (/)c0 3740   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   CCcc 9522   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    < clt 9660    <_ cle 9661    - cmin 9843    / cdiv 10249   NNcn 10578   NN0cn0 10838   ZZcz 10907   ZZ>=cuz 11129   ...cfz 11728   !cfa 12399   prod_cprod 13866   FallFac cfallfac 13951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-rp 11268  df-fz 11729  df-fzo 11857  df-seq 12154  df-exp 12213  df-fac 12400  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-clim 13462  df-prod 13867  df-fallfac 13954
This theorem is referenced by:  bcfallfac  13991  fallfacfac  13992
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