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Mirrors > Home > MPE Home > Th. List > fallfacval2 | Structured version Visualization version GIF version |
Description: One-based value of falling factorial. (Contributed by Scott Fenton, 15-Jan-2018.) |
Ref | Expression |
---|---|
fallfacval2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (1...𝑁)(𝐴 − (𝑘 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fallfacval 14946 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑛 ∈ (0...(𝑁 − 1))(𝐴 − 𝑛)) | |
2 | 1zzd 11610 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 1 ∈ ℤ) | |
3 | 0zd 11591 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 0 ∈ ℤ) | |
4 | nn0z 11602 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
5 | peano2zm 11622 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 1) ∈ ℤ) |
7 | 6 | adantl 467 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑁 − 1) ∈ ℤ) |
8 | simpl 468 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ ℂ) | |
9 | elfznn0 12640 | . . . . 5 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℕ0) | |
10 | 9 | nn0cnd 11555 | . . . 4 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℂ) |
11 | subcl 10482 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝐴 − 𝑛) ∈ ℂ) | |
12 | 8, 10, 11 | syl2an 583 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝐴 − 𝑛) ∈ ℂ) |
13 | oveq2 6801 | . . 3 ⊢ (𝑛 = (𝑘 − 1) → (𝐴 − 𝑛) = (𝐴 − (𝑘 − 1))) | |
14 | 2, 3, 7, 12, 13 | fprodshft 14913 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∏𝑛 ∈ (0...(𝑁 − 1))(𝐴 − 𝑛) = ∏𝑘 ∈ ((0 + 1)...((𝑁 − 1) + 1))(𝐴 − (𝑘 − 1))) |
15 | 0p1e1 11334 | . . . . 5 ⊢ (0 + 1) = 1 | |
16 | 15 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (0 + 1) = 1) |
17 | nn0cn 11504 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
18 | 1cnd 10258 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) | |
19 | 17, 18 | npcand 10598 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) + 1) = 𝑁) |
20 | 19 | adantl 467 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑁 − 1) + 1) = 𝑁) |
21 | 16, 20 | oveq12d 6811 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁)) |
22 | 21 | prodeq1d 14858 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∏𝑘 ∈ ((0 + 1)...((𝑁 − 1) + 1))(𝐴 − (𝑘 − 1)) = ∏𝑘 ∈ (1...𝑁)(𝐴 − (𝑘 − 1))) |
23 | 1, 14, 22 | 3eqtrd 2809 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (1...𝑁)(𝐴 − (𝑘 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 (class class class)co 6793 ℂcc 10136 0cc0 10138 1c1 10139 + caddc 10141 − cmin 10468 ℕ0cn0 11494 ℤcz 11579 ...cfz 12533 ∏cprod 14842 FallFac cfallfac 14941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-prod 14843 df-fallfac 14944 |
This theorem is referenced by: risefallfac 14961 fallfacfwd 14973 |
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