Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fallrisefac Structured version   Unicode version

Theorem fallrisefac 27665
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 10693 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  CC )
212timesd 10671 . . . . . . 7  |-  ( N  e.  NN0  ->  ( 2  x.  N )  =  ( N  +  N
) )
32oveq2d 6209 . . . . . 6  |-  ( N  e.  NN0  ->  ( -u
1 ^ ( 2  x.  N ) )  =  ( -u 1 ^ ( N  +  N ) ) )
4 nn0z 10773 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
5 m1expevenALT 27244 . . . . . . 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 10529 . . . . . . . 8  |-  -u 1  e.  CC
8 expadd 12016 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  N  e.  NN0  /\  N  e.  NN0 )  -> 
( -u 1 ^ ( N  +  N )
)  =  ( (
-u 1 ^ N
)  x.  ( -u
1 ^ N ) ) )
97, 8mp3an1 1302 . . . . . . 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 2501 . . . . 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 9763 . . . . . 6  |-  ( X  e.  CC  ->  -u -u X  =  X )
1413adantr 465 . . . . 5  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  ->  -u -u X  =  X
)
1514oveq1d 6208 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( -u -u X FallFac  N )  =  ( X FallFac  N
) )
1612, 15oveq12d 6211 . . 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 11993 . . . . . 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 9714 . . . . . 6  |-  ( X  e.  CC  ->  -u X  e.  CC )
2120negcld 9810 . . . . 5  |-  ( X  e.  CC  ->  -u -u X  e.  CC )
22 fallfaccl 27656 . . . . 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 9513 . . 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 27656 . . . 4  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( X FallFac  N )  e.  CC )
2625mulid2d 9508 . . 3  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( 1  x.  ( X FallFac  N ) )  =  ( X FallFac  N )
)
2716, 24, 263eqtr3rd 2501 . 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 27664 . . . 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 6209 . 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 2495 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 1370    e. wcel 1758  (class class class)co 6193   CCcc 9384   1c1 9387    + caddc 9389    x. cmul 9391   -ucneg 9700   2c2 10475   NN0cn0 10683   ZZcz 10750   ^cexp 11975   FallFac cfallfac 27644   RiseFac crisefac 27645
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 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-prod 27556  df-risefac 27646  df-fallfac 27647
This theorem is referenced by:  fallfac0  27668
  Copyright terms: Public domain W3C validator