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Theorem risefallfac 25292
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 9262 . . . . . . 7  |-  ( X  e.  CC  ->  -u X  e.  CC )
21adantr 452 . . . . . 6  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  -u X  e.  CC )
3 elfznn 11036 . . . . . . . 8  |-  ( k  e.  ( 1 ... N )  ->  k  e.  NN )
4 nnm1nn0 10217 . . . . . . . 8  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
53, 4syl 16 . . . . . . 7  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  NN0 )
65nn0cnd 10232 . . . . . 6  |-  ( k  e.  ( 1 ... N )  ->  (
k  -  1 )  e.  CC )
7 subcl 9261 . . . . . 6  |-  ( (
-u X  e.  CC  /\  ( k  -  1 )  e.  CC )  ->  ( -u X  -  ( k  - 
1 ) )  e.  CC )
82, 6, 7syl2an 464 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( -u X  -  ( k  - 
1 ) )  e.  CC )
98mulm1d 9441 . . . 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 731 . . . . . 6  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  X  e.  CC )
116adantl 453 . . . . . 6  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( k  - 
1 )  e.  CC )
1210, 11negdi2d 9381 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u ( X  +  ( k  -  1 ) )  =  (
-u X  -  (
k  -  1 ) ) )
1312negeqd 9256 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u -u ( X  +  ( k  -  1 ) )  =  -u ( -u X  -  (
k  -  1 ) ) )
14 simpl 444 . . . . . 6  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  X  e.  CC )
15 addcl 9028 . . . . . 6  |-  ( ( X  e.  CC  /\  ( k  -  1 )  e.  CC )  ->  ( X  +  ( k  -  1 ) )  e.  CC )
1614, 6, 15syl2an 464 . . . . 5  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( X  +  ( k  -  1 ) )  e.  CC )
1716negnegd 9358 . . . 4  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  -u -u ( X  +  ( k  -  1 ) )  =  ( X  +  ( k  -  1 ) ) )
189, 13, 173eqtr2rd 2443 . . 3  |-  ( ( ( X  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
1 ... N ) )  ->  ( X  +  ( k  -  1 ) )  =  (
-u 1  x.  ( -u X  -  ( k  -  1 ) ) ) )
1918prodeq2dv 25202 . 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 25279 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X RiseFac  N )  =  prod_ k  e.  ( 1 ... N ) ( X  +  ( k  -  1 ) ) )
21 fzfi 11266 . . . . . . 7  |-  ( 1 ... N )  e. 
Fin
22 neg1cn 10023 . . . . . . 7  |-  -u 1  e.  CC
23 fprodconst 25255 . . . . . . 7  |-  ( ( ( 1 ... N
)  e.  Fin  /\  -u 1  e.  CC )  ->  prod_ k  e.  ( 1 ... N )
-u 1  =  (
-u 1 ^ ( # `
 ( 1 ... N ) ) ) )
2421, 22, 23mp2an 654 . . . . . 6  |-  prod_ k  e.  ( 1 ... N
) -u 1  =  (
-u 1 ^ ( # `
 ( 1 ... N ) ) )
25 hashfz1 11585 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2625oveq2d 6056 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( # `  ( 1 ... N
) ) )  =  ( -u 1 ^ N ) )
2724, 26syl5req 2449 . . . . 5  |-  ( N  e.  NN0  ->  ( -u
1 ^ N )  =  prod_ k  e.  ( 1 ... N )
-u 1 )
2827adantl 453 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u 1 ^ N
)  =  prod_ k  e.  ( 1 ... N
) -u 1 )
29 fallfacval2 25280 . . . . 5  |-  ( (
-u X  e.  CC  /\  N  e.  NN0 )  ->  ( -u X FallFac  N
)  =  prod_ k  e.  ( 1 ... N
) ( -u X  -  ( k  - 
1 ) ) )
301, 29sylan 458 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u X FallFac  N )  =  prod_ k  e.  ( 1 ... N ) ( -u X  -  ( k  -  1 ) ) )
3128, 30oveq12d 6058 . . 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 11267 . . . 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 25237 . . 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 2439 . 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 2446 1  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X RiseFac  N )  =  ( ( -u
1 ^ N )  x.  ( -u X FallFac  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248   NNcn 9956   NN0cn0 10177   ...cfz 10999   ^cexp 11337   #chash 11573   prod_cprod 25184   FallFac cfallfac 25273   RiseFac crisefac 25274
This theorem is referenced by:  fallrisefac  25293  0risefac  25305  binomrisefac  25309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-prod 25185  df-risefac 25275  df-fallfac 25276
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