![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fallfacval3 | Structured version Visualization version GIF version |
Description: A product representation of falling factorial when 𝐴 is a nonnegative integer. (Contributed by Scott Fenton, 20-Mar-2018.) |
Ref | Expression |
---|---|
fallfacval3 | ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz3nn0 12996 | . . . 4 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℕ0) | |
2 | 1 | nn0cnd 11945 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℂ) |
3 | elfznn0 12995 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → 𝑁 ∈ ℕ0) | |
4 | fallfacval 15355 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑗 ∈ (0...(𝑁 − 1))(𝐴 − 𝑗)) | |
5 | 2, 3, 4 | syl2anc 587 | . 2 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ∏𝑗 ∈ (0...(𝑁 − 1))(𝐴 − 𝑗)) |
6 | elfzel2 12900 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℤ) | |
7 | elfzel1 12901 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → 0 ∈ ℤ) | |
8 | elfzelz 12902 | . . . 4 ⊢ (𝑁 ∈ (0...𝐴) → 𝑁 ∈ ℤ) | |
9 | peano2zm 12013 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → (𝑁 − 1) ∈ ℤ) |
11 | elfzelz 12902 | . . . . 5 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℤ) | |
12 | 11 | zcnd 12076 | . . . 4 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℂ) |
13 | subcl 10874 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝐴 − 𝑗) ∈ ℂ) | |
14 | 2, 12, 13 | syl2an 598 | . . 3 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝐴 − 𝑗) ∈ ℂ) |
15 | oveq2 7143 | . . 3 ⊢ (𝑗 = (𝐴 − 𝑘) → (𝐴 − 𝑗) = (𝐴 − (𝐴 − 𝑘))) | |
16 | 6, 7, 10, 14, 15 | fprodrev 15323 | . 2 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑗 ∈ (0...(𝑁 − 1))(𝐴 − 𝑗) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))(𝐴 − (𝐴 − 𝑘))) |
17 | 2 | subid1d 10975 | . . . 4 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 − 0) = 𝐴) |
18 | 17 | oveq2d 7151 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → ((𝐴 − (𝑁 − 1))...(𝐴 − 0)) = ((𝐴 − (𝑁 − 1))...𝐴)) |
19 | 2 | adantr 484 | . . . 4 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))) → 𝐴 ∈ ℂ) |
20 | elfzelz 12902 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0)) → 𝑘 ∈ ℤ) | |
21 | 20 | zcnd 12076 | . . . . 5 ⊢ (𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0)) → 𝑘 ∈ ℂ) |
22 | 21 | adantl 485 | . . . 4 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))) → 𝑘 ∈ ℂ) |
23 | 19, 22 | nncand 10991 | . . 3 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))) → (𝐴 − (𝐴 − 𝑘)) = 𝑘) |
24 | 18, 23 | prodeq12dv 15272 | . 2 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))(𝐴 − (𝐴 − 𝑘)) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘) |
25 | 5, 16, 24 | 3eqtrd 2837 | 1 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 − cmin 10859 ℕ0cn0 11885 ℤcz 11969 ...cfz 12885 ∏cprod 15251 FallFac cfallfac 15350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-prod 15252 df-fallfac 15353 |
This theorem is referenced by: fallfacval4 15389 |
Copyright terms: Public domain | W3C validator |