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Mirrors > Home > MPE Home > Th. List > 0fallfac | Structured version Visualization version GIF version |
Description: The value of the zero falling factorial at natural 𝑁. (Contributed by Scott Fenton, 17-Feb-2018.) |
Ref | Expression |
---|---|
0fallfac | ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11105 | . . 3 ⊢ 0 ∈ ℂ | |
2 | nnnn0 12378 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | fallfacval 15846 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (0 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘)) | |
4 | 1, 2, 3 | sylancr 587 | . 2 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘)) |
5 | nnm1nn0 12412 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
6 | nn0uz 12759 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | 5, 6 | eleqtrdi 2848 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ≥‘0)) |
8 | elfzelz 13395 | . . . . . 6 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ) | |
9 | 8 | zcnd 12566 | . . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℂ) |
10 | subcl 11358 | . . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (0 − 𝑘) ∈ ℂ) | |
11 | 1, 9, 10 | sylancr 587 | . . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (0 − 𝑘) ∈ ℂ) |
12 | 11 | adantl 482 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (0 − 𝑘) ∈ ℂ) |
13 | oveq2 7359 | . . . 4 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0)) | |
14 | 0m0e0 12231 | . . . 4 ⊢ (0 − 0) = 0 | |
15 | 13, 14 | eqtrdi 2792 | . . 3 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
16 | 7, 12, 15 | fprod1p 15805 | . 2 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘) = (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘))) |
17 | fzfid 13832 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((0 + 1)...(𝑁 − 1)) ∈ Fin) | |
18 | elfzelz 13395 | . . . . . . 7 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℤ) | |
19 | 18 | zcnd 12566 | . . . . . 6 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℂ) |
20 | 1, 19, 10 | sylancr 587 | . . . . 5 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → (0 − 𝑘) ∈ ℂ) |
21 | 20 | adantl 482 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → (0 − 𝑘) ∈ ℂ) |
22 | 17, 21 | fprodcl 15789 | . . 3 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘) ∈ ℂ) |
23 | 22 | mul02d 11311 | . 2 ⊢ (𝑁 ∈ ℕ → (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘)) = 0) |
24 | 4, 16, 23 | 3eqtrd 2780 | 1 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 − cmin 11343 ℕcn 12111 ℕ0cn0 12371 ℤ≥cuz 12721 ...cfz 13378 ∏cprod 15742 FallFac cfallfac 15841 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-fz 13379 df-fzo 13522 df-seq 13861 df-exp 13922 df-hash 14185 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-clim 15324 df-prod 15743 df-fallfac 15844 |
This theorem is referenced by: 0risefac 15875 |
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