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Theorem risefallfac 29114
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 9820 . . . . . . 7  |-  ( X  e.  CC  ->  -u X  e.  CC )
21adantr 465 . . . . . 6  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  -u X  e.  CC )
3 elfznn 11718 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
4 nnm1nn0 10838 . . . . . . . 8  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
53, 4syl 16 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  NN0 )
65nn0cnd 10855 . . . . . 6  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  CC )
7 subcl 9819 . . . . . 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 10009 . . . 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 9945 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u ( X  +  ( k  -  1 ) )  =  (
-u X  -  (
k  -  1 ) ) )
1312negeqd 9814 . . . 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 9572 . . . . . 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 9922 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u -u ( X  +  ( k  -  1 ) )  =  ( X  +  ( k  -  1 ) ) )
189, 13, 173eqtr2rd 2489 . . 3  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( X  +  ( k  -  1 ) )  =  (
-u 1  x.  ( -u X  -  ( k  -  1 ) ) ) )
1918prodeq2dv 29023 . 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 29100 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X RiseFac  N )  =  prod_ k  e.  ( 1 ... N ) ( X  +  ( k  -  1 ) ) )
21 fzfi 12056 . . . . . . 7  |-  ( 1 ... N )  e. 
Fin
22 neg1cn 10640 . . . . . . 7  |-  -u 1  e.  CC
23 fprodconst 29076 . . . . . . 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 12393 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2625oveq2d 6293 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( # `  ( 1 ... N
) ) )  =  ( -u 1 ^ N ) )
2724, 26syl5req 2495 . . . . 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 29101 . . . . 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 6295 . . 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 12057 . . . 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 29058 . . 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 2485 . 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 2492 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 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   Fincfn 7514   CCcc 9488   1c1 9491    + caddc 9493    x. cmul 9495    - cmin 9805   -ucneg 9806   NNcn 10537   NN0cn0 10796   ...cfz 11676   ^cexp 12140   #chash 12379   prod_cprod 29005   FallFac cfallfac 29094   RiseFac crisefac 29095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-prod 29006  df-risefac 29096  df-fallfac 29097
This theorem is referenced by:  fallrisefac  29115  0risefac  29128  binomrisefac  29132
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