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Theorem fallrisefac 27479
Description: A relationship between falling and rising factorials. (Contributed by Scott Fenton, 17-Jan-2018.)
Assertion
Ref Expression
fallrisefac  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X FallFac  N )  =  ( ( -u
1 ^ N )  x.  ( -u X RiseFac  N ) ) )

Proof of Theorem fallrisefac
StepHypRef Expression
1 nn0cn 10581 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
212timesd 10559 . . . . . . 7  |-  ( N  e.  NN0  ->  ( 2  x.  N )  =  ( N  +  N
) )
32oveq2d 6102 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( 2  x.  N ) )  =  ( -u 1 ^ ( N  +  N ) ) )
4 nn0z 10661 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
5 m1expevenALT 27059 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  1 )
64, 5syl 16 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( 2  x.  N ) )  =  1 )
7 neg1cn 10417 . . . . . . . 8  |-  -u 1  e.  CC
8 expadd 11898 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  N  e.  NN0  /\  N  e.  NN0 )  -> 
( -u 1 ^ ( N  +  N )
)  =  ( (
-u 1 ^ N
)  x.  ( -u
1 ^ N ) ) )
97, 8mp3an1 1301 . . . . . . 7  |-  ( ( N  e.  NN0  /\  N  e.  NN0 )  -> 
( -u 1 ^ ( N  +  N )
)  =  ( (
-u 1 ^ N
)  x.  ( -u
1 ^ N ) ) )
109anidms 645 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( N  +  N ) )  =  ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) ) )
113, 6, 103eqtr3rd 2479 . . . . 5  |-  ( N  e.  NN0  ->  ( (
-u 1 ^ N
)  x.  ( -u
1 ^ N ) )  =  1 )
1211adantl 466 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
13 negneg 9651 . . . . . 6  |-  ( X  e.  CC  ->  -u -u X  =  X )
1413adantr 465 . . . . 5  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  -u -u X  =  X
)
1514oveq1d 6101 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u -u X FallFac  N )  =  ( X FallFac  N
) )
1612, 15oveq12d 6104 . . 3  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) )  x.  ( -u -u X FallFac  N ) )  =  ( 1  x.  ( X FallFac  N ) ) )
17 expcl 11875 . . . . . 6  |-  ( (
-u 1  e.  CC  /\  N  e.  NN0 )  ->  ( -u 1 ^ N )  e.  CC )
187, 17mpan 670 . . . . 5  |-  ( N  e.  NN0  ->  ( -u
1 ^ N )  e.  CC )
1918adantl 466 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u 1 ^ N
)  e.  CC )
20 negcl 9602 . . . . . 6  |-  ( X  e.  CC  ->  -u X  e.  CC )
2120negcld 9698 . . . . 5  |-  ( X  e.  CC  ->  -u -u X  e.  CC )
22 fallfaccl 27470 . . . . 5  |-  ( (
-u -u X  e.  CC  /\  N  e.  NN0 )  ->  ( -u -u X FallFac  N )  e.  CC )
2321, 22sylan 471 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u -u X FallFac  N )  e.  CC )
2419, 19, 23mulassd 9401 . . 3  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) )  x.  ( -u -u X FallFac  N ) )  =  ( ( -u 1 ^ N )  x.  (
( -u 1 ^ N
)  x.  ( -u -u X FallFac  N ) ) ) )
25 fallfaccl 27470 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X FallFac  N )  e.  CC )
2625mulid2d 9396 . . 3  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( 1  x.  ( X FallFac  N ) )  =  ( X FallFac  N )
)
2716, 24, 263eqtr3rd 2479 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X FallFac  N )  =  ( ( -u
1 ^ N )  x.  ( ( -u
1 ^ N )  x.  ( -u -u X FallFac  N ) ) ) )
28 risefallfac 27478 . . . 4  |-  ( (
-u X  e.  CC  /\  N  e.  NN0 )  ->  ( -u X RiseFac  N
)  =  ( (
-u 1 ^ N
)  x.  ( -u -u X FallFac  N ) ) )
2920, 28sylan 471 . . 3  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u X RiseFac  N )  =  ( ( -u
1 ^ N )  x.  ( -u -u X FallFac  N ) ) )
3029oveq2d 6102 . 2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( ( -u 1 ^ N )  x.  ( -u X RiseFac  N ) )  =  ( ( -u 1 ^ N )  x.  (
( -u 1 ^ N
)  x.  ( -u -u X FallFac  N ) ) ) )
3127, 30eqtr4d 2473 1  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X FallFac  N )  =  ( ( -u
1 ^ N )  x.  ( -u X RiseFac  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756  (class class class)co 6086   CCcc 9272   1c1 9275    + caddc 9277    x. cmul 9279   -ucneg 9588   2c2 10363   NN0cn0 10571   ZZcz 10638   ^cexp 11857   FallFac cfallfac 27458   RiseFac crisefac 27459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-prod 27370  df-risefac 27460  df-fallfac 27461
This theorem is referenced by:  fallfac0  27482
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