![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 15897 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑛 ∈ (0...(𝑁 − 1))(𝐴 − 𝑛)) | |
2 | 1zzd 12539 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 1 ∈ ℤ) | |
3 | 0zd 12516 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 0 ∈ ℤ) | |
4 | nn0z 12529 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
5 | peano2zm 12551 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 1) ∈ ℤ) |
7 | 6 | adantl 483 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝑁 − 1) ∈ ℤ) |
8 | simpl 484 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ ℂ) | |
9 | elfznn0 13540 | . . . . 5 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℕ0) | |
10 | 9 | nn0cnd 12480 | . . . 4 ⊢ (𝑛 ∈ (0...(𝑁 − 1)) → 𝑛 ∈ ℂ) |
11 | subcl 11405 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝐴 − 𝑛) ∈ ℂ) | |
12 | 8, 10, 11 | syl2an 597 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) ∧ 𝑛 ∈ (0...(𝑁 − 1))) → (𝐴 − 𝑛) ∈ ℂ) |
13 | oveq2 7366 | . . 3 ⊢ (𝑛 = (𝑘 − 1) → (𝐴 − 𝑛) = (𝐴 − (𝑘 − 1))) | |
14 | 2, 3, 7, 12, 13 | fprodshft 15864 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ∏𝑛 ∈ (0...(𝑁 − 1))(𝐴 − 𝑛) = ∏𝑘 ∈ ((0 + 1)...((𝑁 − 1) + 1))(𝐴 − (𝑘 − 1))) |
15 | 0p1e1 12280 | . . . . 5 ⊢ (0 + 1) = 1 | |
16 | 15 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (0 + 1) = 1) |
17 | nn0cn 12428 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
18 | 1cnd 11155 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) | |
19 | 17, 18 | npcand 11521 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) + 1) = 𝑁) |
20 | 19 | adantl 483 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝑁 − 1) + 1) = 𝑁) |
21 | 16, 20 | oveq12d 7376 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁)) |
22 | 21 | prodeq1d 15809 | . 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 2777 | 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 397 = wceq 1542 ∈ wcel 2107 (class class class)co 7358 ℂcc 11054 0cc0 11056 1c1 11057 + caddc 11059 − cmin 11390 ℕ0cn0 12418 ℤcz 12504 ...cfz 13430 ∏cprod 15793 FallFac cfallfac 15892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-prod 15794 df-fallfac 15895 |
This theorem is referenced by: risefallfac 15912 fallfacfwd 15924 |
Copyright terms: Public domain | W3C validator |