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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 10633 | . . 3 ⊢ 0 ∈ ℂ | |
2 | nnnn0 11905 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
3 | fallfacval 15363 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (0 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘)) | |
4 | 1, 2, 3 | sylancr 589 | . 2 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘)) |
5 | nnm1nn0 11939 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
6 | nn0uz 12281 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
7 | 5, 6 | eleqtrdi 2923 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (ℤ≥‘0)) |
8 | elfzelz 12909 | . . . . . 6 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℤ) | |
9 | 8 | zcnd 12089 | . . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℂ) |
10 | subcl 10885 | . . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (0 − 𝑘) ∈ ℂ) | |
11 | 1, 9, 10 | sylancr 589 | . . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (0 − 𝑘) ∈ ℂ) |
12 | 11 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (0 − 𝑘) ∈ ℂ) |
13 | oveq2 7164 | . . . 4 ⊢ (𝑘 = 0 → (0 − 𝑘) = (0 − 0)) | |
14 | 0m0e0 11758 | . . . 4 ⊢ (0 − 0) = 0 | |
15 | 13, 14 | syl6eq 2872 | . . 3 ⊢ (𝑘 = 0 → (0 − 𝑘) = 0) |
16 | 7, 12, 15 | fprod1p 15322 | . 2 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (0...(𝑁 − 1))(0 − 𝑘) = (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘))) |
17 | fzfid 13342 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((0 + 1)...(𝑁 − 1)) ∈ Fin) | |
18 | elfzelz 12909 | . . . . . . 7 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℤ) | |
19 | 18 | zcnd 12089 | . . . . . 6 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → 𝑘 ∈ ℂ) |
20 | 1, 19, 10 | sylancr 589 | . . . . 5 ⊢ (𝑘 ∈ ((0 + 1)...(𝑁 − 1)) → (0 − 𝑘) ∈ ℂ) |
21 | 20 | adantl 484 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ((0 + 1)...(𝑁 − 1))) → (0 − 𝑘) ∈ ℂ) |
22 | 17, 21 | fprodcl 15306 | . . 3 ⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘) ∈ ℂ) |
23 | 22 | mul02d 10838 | . 2 ⊢ (𝑁 ∈ ℕ → (0 · ∏𝑘 ∈ ((0 + 1)...(𝑁 − 1))(0 − 𝑘)) = 0) |
24 | 4, 16, 23 | 3eqtrd 2860 | 1 ⊢ (𝑁 ∈ ℕ → (0 FallFac 𝑁) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 ℕcn 11638 ℕ0cn0 11898 ℤ≥cuz 12244 ...cfz 12893 ∏cprod 15259 FallFac cfallfac 15358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-prod 15260 df-fallfac 15361 |
This theorem is referenced by: 0risefac 15392 |
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