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Theorem risefallfac 27374
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 9598 . . . . . . 7  |-  ( X  e.  CC  ->  -u X  e.  CC )
21adantr 462 . . . . . 6  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  -u X  e.  CC )
3 elfznn 11465 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
4 nnm1nn0 10609 . . . . . . . 8  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
53, 4syl 16 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  NN0 )
65nn0cnd 10626 . . . . . 6  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  CC )
7 subcl 9597 . . . . . 6  |-  ( (
-u X  e.  CC  /\  ( k  -  1 )  e.  CC )  ->  ( -u X  -  ( k  - 
1 ) )  e.  CC )
82, 6, 7syl2an 474 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( -u X  -  ( k  - 
1 ) )  e.  CC )
98mulm1d 9784 . . . 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 746 . . . . . 6  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  X  e.  CC )
116adantl 463 . . . . . 6  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( k  - 
1 )  e.  CC )
1210, 11negdi2d 9721 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u ( X  +  ( k  -  1 ) )  =  (
-u X  -  (
k  -  1 ) ) )
1312negeqd 9592 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u -u ( X  +  ( k  -  1 ) )  =  -u ( -u X  -  (
k  -  1 ) ) )
14 simpl 454 . . . . . 6  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  X  e.  CC )
15 addcl 9352 . . . . . 6  |-  ( ( X  e.  CC  /\  ( k  -  1 )  e.  CC )  ->  ( X  +  ( k  -  1 ) )  e.  CC )
1614, 6, 15syl2an 474 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( X  +  ( k  -  1 ) )  e.  CC )
1716negnegd 9698 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u -u ( X  +  ( k  -  1 ) )  =  ( X  +  ( k  -  1 ) ) )
189, 13, 173eqtr2rd 2472 . . 3  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( X  +  ( k  -  1 ) )  =  (
-u 1  x.  ( -u X  -  ( k  -  1 ) ) ) )
1918prodeq2dv 27283 . 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 27360 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X RiseFac  N )  =  prod_ k  e.  ( 1 ... N ) ( X  +  ( k  -  1 ) ) )
21 fzfi 11778 . . . . . . 7  |-  ( 1 ... N )  e. 
Fin
22 neg1cn 10413 . . . . . . 7  |-  -u 1  e.  CC
23 fprodconst 27336 . . . . . . 7  |-  ( ( ( 1 ... N
)  e.  Fin  /\  -u 1  e.  CC )  ->  prod_ k  e.  ( 1 ... N )
-u 1  =  (
-u 1 ^ ( # `
 ( 1 ... N ) ) ) )
2421, 22, 23mp2an 665 . . . . . 6  |-  prod_ k  e.  ( 1 ... N
) -u 1  =  (
-u 1 ^ ( # `
 ( 1 ... N ) ) )
25 hashfz1 12101 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2625oveq2d 6096 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( # `  ( 1 ... N
) ) )  =  ( -u 1 ^ N ) )
2724, 26syl5req 2478 . . . . 5  |-  ( N  e.  NN0  ->  ( -u
1 ^ N )  =  prod_ k  e.  ( 1 ... N )
-u 1 )
2827adantl 463 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u 1 ^ N
)  =  prod_ k  e.  ( 1 ... N
) -u 1 )
29 fallfacval2 27361 . . . . 5  |-  ( (
-u X  e.  CC  /\  N  e.  NN0 )  ->  ( -u X FallFac  N
)  =  prod_ k  e.  ( 1 ... N
) ( -u X  -  ( k  - 
1 ) ) )
301, 29sylan 468 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u X FallFac  N )  =  prod_ k  e.  ( 1 ... N ) ( -u X  -  ( k  -  1 ) ) )
3128, 30oveq12d 6098 . . 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 11779 . . . 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 27318 . . 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 2468 . 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 2475 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 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9268   1c1 9271    + caddc 9273    x. cmul 9275    - cmin 9583   -ucneg 9584   NNcn 10310   NN0cn0 10567   ...cfz 11424   ^cexp 11849   #chash 12087   prod_cprod 27265   FallFac cfallfac 27354   RiseFac crisefac 27355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-prod 27266  df-risefac 27356  df-fallfac 27357
This theorem is referenced by:  fallrisefac  27375  0risefac  27388  binomrisefac  27392
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