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Theorem fallfacval4 27682
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 11898 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
( ( A  -  N )  +  1 ) ... A )  e.  Fin )
2 elfzelz 11556 . . . . . . 7  |-  ( k  e.  ( ( ( A  -  N )  +  1 ) ... A )  ->  k  e.  ZZ )
32zcnd 10851 . . . . . 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 27601 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( ( ( A  -  N )  +  1 ) ... A ) k  e.  CC )
6 fzfid 11898 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
1 ... ( A  -  N ) )  e. 
Fin )
7 elfznn 11581 . . . . . . 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 10441 . . . . 5  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( 1 ... ( A  -  N ) ) )  ->  k  e.  CC )
106, 9fprodcl 27601 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( 1 ... ( A  -  N
) ) k  e.  CC )
118nnne0d 10469 . . . . 5  |-  ( ( N  e.  ( 0 ... A )  /\  k  e.  ( 1 ... ( A  -  N ) ) )  ->  k  =/=  0
)
126, 9, 11fprodn0 27626 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( 1 ... ( A  -  N
) ) k  =/=  0 )
135, 10, 12divcan3d 10215 . . 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 11590 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  NN0 )
1514nn0red 10740 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  RR )
1615ltp1d 10366 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  <  ( ( A  -  N )  +  1 ) )
17 fzdisj 11579 . . . . . 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 10722 . . . . . . . 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 10999 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2220, 21syl6eleq 2549 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  (
( A  -  N
)  +  1 )  e.  ( ZZ>= `  1
) )
2314nn0zd 10848 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  e.  ZZ )
24 elfzel2 11554 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  A  e.  ZZ )
25 elfzle1 11557 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  0  <_  N )
2624zred 10850 . . . . . . . . 9  |-  ( N  e.  ( 0 ... A )  ->  A  e.  RR )
27 elfzelz 11556 . . . . . . . . . 10  |-  ( N  e.  ( 0 ... A )  ->  N  e.  ZZ )
2827zred 10850 . . . . . . . . 9  |-  ( N  e.  ( 0 ... A )  ->  N  e.  RR )
2926, 28subge02d 10034 . . . . . . . 8  |-  ( N  e.  ( 0 ... A )  ->  (
0  <_  N  <->  ( A  -  N )  <_  A
) )
3025, 29mpbid 210 . . . . . . 7  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  N )  <_  A )
31 eluz2 10970 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  ( A  -  N )
)  <->  ( ( A  -  N )  e.  ZZ  /\  A  e.  ZZ  /\  ( A  -  N )  <_  A ) )
3223, 24, 30, 31syl3anbrc 1172 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  A  e.  ( ZZ>= `  ( A  -  N ) ) )
33 fzsplit2 11577 . . . . . 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 11898 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  (
1 ... A )  e. 
Fin )
36 elfznn 11581 . . . . . . 7  |-  ( k  e.  ( 1 ... A )  ->  k  e.  NN )
3736nncnd 10441 . . . . . 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 27612 . . . 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 6207 . . 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 10851 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  A  e.  CC )
4227zcnd 10851 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  N  e.  CC )
43 ax-1cn 9443 . . . . . . 7  |-  1  e.  CC
4443a1i 11 . . . . . 6  |-  ( N  e.  ( 0 ... A )  ->  1  e.  CC )
4541, 42, 44subsubd 9850 . . . . 5  |-  ( N  e.  ( 0 ... A )  ->  ( A  -  ( N  -  1 ) )  =  ( ( A  -  N )  +  1 ) )
4645oveq1d 6207 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  (
( A  -  ( N  -  1 ) ) ... A )  =  ( ( ( A  -  N )  +  1 ) ... A ) )
4746prodeq1d 27570 . . 3  |-  ( N  e.  ( 0 ... A )  ->  prod_ k  e.  ( ( A  -  ( N  - 
1 ) ) ... A ) k  = 
prod_ k  e.  (
( ( A  -  N )  +  1 ) ... A ) k )
4813, 40, 473eqtr4rd 2503 . 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 27651 . 2  |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  prod_ k  e.  ( ( A  -  ( N  - 
1 ) ) ... A ) k )
50 elfz3nn0 11585 . . . 4  |-  ( N  e.  ( 0 ... A )  ->  A  e.  NN0 )
51 fprodfac 27619 . . . 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 27619 . . . 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 6210 . 2  |-  ( N  e.  ( 0 ... A )  ->  (
( ! `  A
)  /  ( ! `
 ( A  -  N ) ) )  =  ( prod_ k  e.  ( 1 ... A
) k  /  prod_ k  e.  ( 1 ... ( A  -  N
) ) k ) )
5648, 49, 553eqtr4d 2502 1  |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  ( ( ! `  A
)  /  ( ! `
 ( A  -  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3426    i^i cin 3427   (/)c0 3737   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   CCcc 9383   0cc0 9385   1c1 9386    + caddc 9388    x. cmul 9390    < clt 9521    <_ cle 9522    - cmin 9698    / cdiv 10096   NNcn 10425   NN0cn0 10682   ZZcz 10749   ZZ>=cuz 10964   ...cfz 11540   !cfa 12154   prod_cprod 27554   FallFac cfallfac 27643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-fz 11541  df-fzo 11652  df-seq 11910  df-exp 11969  df-fac 12155  df-hash 12207  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-clim 13070  df-prod 27555  df-fallfac 27646
This theorem is referenced by:  bcfallfac  27683  fallfacfac  27684
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