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Theorem risefallfac 27694
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 9725 . . . . . . 7  |-  ( X  e.  CC  ->  -u X  e.  CC )
21adantr 465 . . . . . 6  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  -u X  e.  CC )
3 elfznn 11599 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
4 nnm1nn0 10736 . . . . . . . 8  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
53, 4syl 16 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  NN0 )
65nn0cnd 10753 . . . . . 6  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  CC )
7 subcl 9724 . . . . . 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 9911 . . . 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 9848 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u ( X  +  ( k  -  1 ) )  =  (
-u X  -  (
k  -  1 ) ) )
1312negeqd 9719 . . . 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 9479 . . . . . 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 9825 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u -u ( X  +  ( k  -  1 ) )  =  ( X  +  ( k  -  1 ) ) )
189, 13, 173eqtr2rd 2502 . . 3  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( X  +  ( k  -  1 ) )  =  (
-u 1  x.  ( -u X  -  ( k  -  1 ) ) ) )
1918prodeq2dv 27603 . 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 27680 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X RiseFac  N )  =  prod_ k  e.  ( 1 ... N ) ( X  +  ( k  -  1 ) ) )
21 fzfi 11915 . . . . . . 7  |-  ( 1 ... N )  e. 
Fin
22 neg1cn 10540 . . . . . . 7  |-  -u 1  e.  CC
23 fprodconst 27656 . . . . . . 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 12238 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2625oveq2d 6219 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( # `  ( 1 ... N
) ) )  =  ( -u 1 ^ N ) )
2724, 26syl5req 2508 . . . . 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 27681 . . . . 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 6221 . . 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 11916 . . . 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 27638 . . 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 2498 . 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 2505 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 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   Fincfn 7423   CCcc 9395   1c1 9398    + caddc 9400    x. cmul 9402    - cmin 9710   -ucneg 9711   NNcn 10437   NN0cn0 10694   ...cfz 11558   ^cexp 11986   #chash 12224   prod_cprod 27585   FallFac cfallfac 27674   RiseFac crisefac 27675
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fz 11559  df-fzo 11670  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-prod 27586  df-risefac 27676  df-fallfac 27677
This theorem is referenced by:  fallrisefac  27695  0risefac  27708  binomrisefac  27712
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