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Mirrors > Home > MPE Home > Th. List > risefallfac | Structured version Visualization version GIF version |
Description: A relationship between rising and falling factorials. (Contributed by Scott Fenton, 15-Jan-2018.) |
Ref | Expression |
---|---|
risefallfac | ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑋 RiseFac 𝑁) = ((-1↑𝑁) · (-𝑋 FallFac 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 10885 | . . . . . . 7 ⊢ (𝑋 ∈ ℂ → -𝑋 ∈ ℂ) | |
2 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → -𝑋 ∈ ℂ) |
3 | elfznn 12935 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
4 | nnm1nn0 11937 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ0) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ (1...𝑁) → (𝑘 − 1) ∈ ℕ0) |
6 | 5 | nn0cnd 11956 | . . . . . 6 ⊢ (𝑘 ∈ (1...𝑁) → (𝑘 − 1) ∈ ℂ) |
7 | subcl 10884 | . . . . . 6 ⊢ ((-𝑋 ∈ ℂ ∧ (𝑘 − 1) ∈ ℂ) → (-𝑋 − (𝑘 − 1)) ∈ ℂ) | |
8 | 2, 6, 7 | syl2an 597 | . . . . 5 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → (-𝑋 − (𝑘 − 1)) ∈ ℂ) |
9 | 8 | mulm1d 11091 | . . . 4 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → (-1 · (-𝑋 − (𝑘 − 1))) = -(-𝑋 − (𝑘 − 1))) |
10 | simpll 765 | . . . . . 6 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → 𝑋 ∈ ℂ) | |
11 | 6 | adantl 484 | . . . . . 6 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 − 1) ∈ ℂ) |
12 | 10, 11 | negdi2d 11010 | . . . . 5 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → -(𝑋 + (𝑘 − 1)) = (-𝑋 − (𝑘 − 1))) |
13 | 12 | negeqd 10879 | . . . 4 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → --(𝑋 + (𝑘 − 1)) = -(-𝑋 − (𝑘 − 1))) |
14 | simpl 485 | . . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝑋 ∈ ℂ) | |
15 | addcl 10618 | . . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 − 1) ∈ ℂ) → (𝑋 + (𝑘 − 1)) ∈ ℂ) | |
16 | 14, 6, 15 | syl2an 597 | . . . . 5 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → (𝑋 + (𝑘 − 1)) ∈ ℂ) |
17 | 16 | negnegd 10987 | . . . 4 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → --(𝑋 + (𝑘 − 1)) = (𝑋 + (𝑘 − 1))) |
18 | 9, 13, 17 | 3eqtr2rd 2863 | . . 3 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → (𝑋 + (𝑘 − 1)) = (-1 · (-𝑋 − (𝑘 − 1)))) |
19 | 18 | prodeq2dv 15276 | . 2 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∏𝑘 ∈ (1...𝑁)(𝑋 + (𝑘 − 1)) = ∏𝑘 ∈ (1...𝑁)(-1 · (-𝑋 − (𝑘 − 1)))) |
20 | risefacval2 15363 | . 2 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑋 RiseFac 𝑁) = ∏𝑘 ∈ (1...𝑁)(𝑋 + (𝑘 − 1))) | |
21 | fzfi 13339 | . . . . . . 7 ⊢ (1...𝑁) ∈ Fin | |
22 | neg1cn 11750 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
23 | fprodconst 15331 | . . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ -1 ∈ ℂ) → ∏𝑘 ∈ (1...𝑁)-1 = (-1↑(♯‘(1...𝑁)))) | |
24 | 21, 22, 23 | mp2an 690 | . . . . . 6 ⊢ ∏𝑘 ∈ (1...𝑁)-1 = (-1↑(♯‘(1...𝑁))) |
25 | hashfz1 13705 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁) | |
26 | 25 | oveq2d 7171 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (-1↑(♯‘(1...𝑁))) = (-1↑𝑁)) |
27 | 24, 26 | syl5req 2869 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (-1↑𝑁) = ∏𝑘 ∈ (1...𝑁)-1) |
28 | 27 | adantl 484 | . . . 4 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (-1↑𝑁) = ∏𝑘 ∈ (1...𝑁)-1) |
29 | fallfacval2 15364 | . . . . 5 ⊢ ((-𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (-𝑋 FallFac 𝑁) = ∏𝑘 ∈ (1...𝑁)(-𝑋 − (𝑘 − 1))) | |
30 | 1, 29 | sylan 582 | . . . 4 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (-𝑋 FallFac 𝑁) = ∏𝑘 ∈ (1...𝑁)(-𝑋 − (𝑘 − 1))) |
31 | 28, 30 | oveq12d 7173 | . . 3 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((-1↑𝑁) · (-𝑋 FallFac 𝑁)) = (∏𝑘 ∈ (1...𝑁)-1 · ∏𝑘 ∈ (1...𝑁)(-𝑋 − (𝑘 − 1)))) |
32 | fzfid 13340 | . . . 4 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (1...𝑁) ∈ Fin) | |
33 | 22 | a1i 11 | . . . 4 ⊢ (((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ (1...𝑁)) → -1 ∈ ℂ) |
34 | 32, 33, 8 | fprodmul 15313 | . . 3 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∏𝑘 ∈ (1...𝑁)(-1 · (-𝑋 − (𝑘 − 1))) = (∏𝑘 ∈ (1...𝑁)-1 · ∏𝑘 ∈ (1...𝑁)(-𝑋 − (𝑘 − 1)))) |
35 | 31, 34 | eqtr4d 2859 | . 2 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((-1↑𝑁) · (-𝑋 FallFac 𝑁)) = ∏𝑘 ∈ (1...𝑁)(-1 · (-𝑋 − (𝑘 − 1)))) |
36 | 19, 20, 35 | 3eqtr4d 2866 | 1 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑋 RiseFac 𝑁) = ((-1↑𝑁) · (-𝑋 FallFac 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Fincfn 8508 ℂcc 10534 1c1 10537 + caddc 10539 · cmul 10541 − cmin 10869 -cneg 10870 ℕcn 11637 ℕ0cn0 11896 ...cfz 12891 ↑cexp 13428 ♯chash 13689 ∏cprod 15258 FallFac cfallfac 15357 RiseFac crisefac 15358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-prod 15259 df-risefac 15359 df-fallfac 15360 |
This theorem is referenced by: fallrisefac 15378 0risefac 15391 binomrisefac 15395 |
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