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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 13630 | . . . 4 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℕ0) | |
2 | 1 | nn0cnd 12567 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℂ) |
3 | elfznn0 13629 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → 𝑁 ∈ ℕ0) | |
4 | fallfacval 15989 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑗 ∈ (0...(𝑁 − 1))(𝐴 − 𝑗)) | |
5 | 2, 3, 4 | syl2anc 582 | . 2 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ∏𝑗 ∈ (0...(𝑁 − 1))(𝐴 − 𝑗)) |
6 | elfzel2 13534 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → 𝐴 ∈ ℤ) | |
7 | elfzel1 13535 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → 0 ∈ ℤ) | |
8 | elfzelz 13536 | . . . 4 ⊢ (𝑁 ∈ (0...𝐴) → 𝑁 ∈ ℤ) | |
9 | peano2zm 12638 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → (𝑁 − 1) ∈ ℤ) |
11 | elfzelz 13536 | . . . . 5 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℤ) | |
12 | 11 | zcnd 12700 | . . . 4 ⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℂ) |
13 | subcl 11491 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝐴 − 𝑗) ∈ ℂ) | |
14 | 2, 12, 13 | syl2an 594 | . . 3 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝐴 − 𝑗) ∈ ℂ) |
15 | oveq2 7427 | . . 3 ⊢ (𝑗 = (𝐴 − 𝑘) → (𝐴 − 𝑗) = (𝐴 − (𝐴 − 𝑘))) | |
16 | 6, 7, 10, 14, 15 | fprodrev 15957 | . 2 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑗 ∈ (0...(𝑁 − 1))(𝐴 − 𝑗) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))(𝐴 − (𝐴 − 𝑘))) |
17 | 2 | subid1d 11592 | . . . 4 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 − 0) = 𝐴) |
18 | 17 | oveq2d 7435 | . . 3 ⊢ (𝑁 ∈ (0...𝐴) → ((𝐴 − (𝑁 − 1))...(𝐴 − 0)) = ((𝐴 − (𝑁 − 1))...𝐴)) |
19 | 2 | adantr 479 | . . . 4 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))) → 𝐴 ∈ ℂ) |
20 | elfzelz 13536 | . . . . . 6 ⊢ (𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0)) → 𝑘 ∈ ℤ) | |
21 | 20 | zcnd 12700 | . . . . 5 ⊢ (𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0)) → 𝑘 ∈ ℂ) |
22 | 21 | adantl 480 | . . . 4 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))) → 𝑘 ∈ ℂ) |
23 | 19, 22 | nncand 11608 | . . 3 ⊢ ((𝑁 ∈ (0...𝐴) ∧ 𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))) → (𝐴 − (𝐴 − 𝑘)) = 𝑘) |
24 | 18, 23 | prodeq12dv 15906 | . 2 ⊢ (𝑁 ∈ (0...𝐴) → ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...(𝐴 − 0))(𝐴 − (𝐴 − 𝑘)) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘) |
25 | 5, 16, 24 | 3eqtrd 2769 | 1 ⊢ (𝑁 ∈ (0...𝐴) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ ((𝐴 − (𝑁 − 1))...𝐴)𝑘) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 − cmin 11476 ℕ0cn0 12505 ℤcz 12591 ...cfz 13519 ∏cprod 15885 FallFac cfallfac 15984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-prod 15886 df-fallfac 15987 |
This theorem is referenced by: fallfacval4 16023 |
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