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

Theorem risefallfac 29073
Description: A relationship between rising and falling factorials. (Contributed by Scott Fenton, 15-Jan-2018.)
Assertion
Ref Expression
risefallfac  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X RiseFac  N )  =  ( ( -u
1 ^ N )  x.  ( -u X FallFac  N ) ) )

Proof of Theorem risefallfac
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 negcl 9832 . . . . . . 7  |-  ( X  e.  CC  ->  -u X  e.  CC )
21adantr 465 . . . . . 6  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  -u X  e.  CC )
3 elfznn 11726 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
4 nnm1nn0 10849 . . . . . . . 8  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
53, 4syl 16 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  NN0 )
65nn0cnd 10866 . . . . . 6  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  CC )
7 subcl 9831 . . . . . 6  |-  ( (
-u X  e.  CC  /\  ( k  -  1 )  e.  CC )  ->  ( -u X  -  ( k  - 
1 ) )  e.  CC )
82, 6, 7syl2an 477 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( -u X  -  ( k  - 
1 ) )  e.  CC )
98mulm1d 10020 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( -u 1  x.  ( -u X  -  ( k  -  1 ) ) )  = 
-u ( -u X  -  ( k  - 
1 ) ) )
10 simpll 753 . . . . . 6  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  X  e.  CC )
116adantl 466 . . . . . 6  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( k  - 
1 )  e.  CC )
1210, 11negdi2d 9956 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u ( X  +  ( k  -  1 ) )  =  (
-u X  -  (
k  -  1 ) ) )
1312negeqd 9826 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u -u ( X  +  ( k  -  1 ) )  =  -u ( -u X  -  (
k  -  1 ) ) )
14 simpl 457 . . . . . 6  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  X  e.  CC )
15 addcl 9586 . . . . . 6  |-  ( ( X  e.  CC  /\  ( k  -  1 )  e.  CC )  ->  ( X  +  ( k  -  1 ) )  e.  CC )
1614, 6, 15syl2an 477 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( X  +  ( k  -  1 ) )  e.  CC )
1716negnegd 9933 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u -u ( X  +  ( k  -  1 ) )  =  ( X  +  ( k  -  1 ) ) )
189, 13, 173eqtr2rd 2515 . . 3  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( X  +  ( k  -  1 ) )  =  (
-u 1  x.  ( -u X  -  ( k  -  1 ) ) ) )
1918prodeq2dv 28982 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  prod_ k  e.  ( 1 ... N ) ( X  +  ( k  -  1 ) )  =  prod_ k  e.  ( 1 ... N ) ( -u 1  x.  ( -u X  -  ( k  -  1 ) ) ) )
20 risefacval2 29059 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X RiseFac  N )  =  prod_ k  e.  ( 1 ... N ) ( X  +  ( k  -  1 ) ) )
21 fzfi 12062 . . . . . . 7  |-  ( 1 ... N )  e. 
Fin
22 neg1cn 10651 . . . . . . 7  |-  -u 1  e.  CC
23 fprodconst 29035 . . . . . . 7  |-  ( ( ( 1 ... N
)  e.  Fin  /\  -u 1  e.  CC )  ->  prod_ k  e.  ( 1 ... N )
-u 1  =  (
-u 1 ^ ( # `
 ( 1 ... N ) ) ) )
2421, 22, 23mp2an 672 . . . . . 6  |-  prod_ k  e.  ( 1 ... N
) -u 1  =  (
-u 1 ^ ( # `
 ( 1 ... N ) ) )
25 hashfz1 12399 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2625oveq2d 6311 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( # `  ( 1 ... N
) ) )  =  ( -u 1 ^ N ) )
2724, 26syl5req 2521 . . . . 5  |-  ( N  e.  NN0  ->  ( -u
1 ^ N )  =  prod_ k  e.  ( 1 ... N )
-u 1 )
2827adantl 466 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u 1 ^ N
)  =  prod_ k  e.  ( 1 ... N
) -u 1 )
29 fallfacval2 29060 . . . . 5  |-  ( (
-u X  e.  CC  /\  N  e.  NN0 )  ->  ( -u X FallFac  N
)  =  prod_ k  e.  ( 1 ... N
) ( -u X  -  ( k  - 
1 ) ) )
301, 29sylan 471 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u X FallFac  N )  =  prod_ k  e.  ( 1 ... N ) ( -u X  -  ( k  -  1 ) ) )
3128, 30oveq12d 6313 . . 3  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( ( -u 1 ^ N )  x.  ( -u X FallFac  N ) )  =  ( prod_ k  e.  ( 1 ... N )
-u 1  x.  prod_ k  e.  ( 1 ... N ) ( -u X  -  ( k  -  1 ) ) ) )
32 fzfid 12063 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( 1 ... N
)  e.  Fin )
3322a1i 11 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u 1  e.  CC )
3432, 33, 8fprodmul 29017 . . 3  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  prod_ k  e.  ( 1 ... N ) (
-u 1  x.  ( -u X  -  ( k  -  1 ) ) )  =  ( prod_
k  e.  ( 1 ... N ) -u
1  x.  prod_ k  e.  ( 1 ... N
) ( -u X  -  ( k  - 
1 ) ) ) )
3531, 34eqtr4d 2511 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( ( -u 1 ^ N )  x.  ( -u X FallFac  N ) )  = 
prod_ k  e.  (
1 ... N ) (
-u 1  x.  ( -u X  -  ( k  -  1 ) ) ) )
3619, 20, 353eqtr4d 2518 1  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X RiseFac  N )  =  ( ( -u
1 ^ N )  x.  ( -u X FallFac  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   1c1 9505    + caddc 9507    x. cmul 9509    - cmin 9817   -ucneg 9818   NNcn 10548   NN0cn0 10807   ...cfz 11684   ^cexp 12146   #chash 12385   prod_cprod 28964   FallFac cfallfac 29053   RiseFac crisefac 29054
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-prod 28965  df-risefac 29055  df-fallfac 29056
This theorem is referenced by:  fallrisefac  29074  0risefac  29087  binomrisefac  29091
  Copyright terms: Public domain W3C validator