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| Mirrors > Home > MPE Home > Th. List > fallfacval | Structured version Visualization version GIF version | ||
| Description: The value of the falling factorial function. (Contributed by Scott Fenton, 5-Jan-2018.) |
| Ref | Expression |
|---|---|
| fallfacval | ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − 𝑘)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq1 7438 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 − 𝑘) = (𝐴 − 𝑘)) | |
| 2 | 1 | prodeq2sdv 15959 | . 2 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘) = ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 − 𝑘)) |
| 3 | oveq1 7438 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1)) | |
| 4 | 3 | oveq2d 7447 | . . 3 ⊢ (𝑛 = 𝑁 → (0...(𝑛 − 1)) = (0...(𝑁 − 1))) |
| 5 | 4 | prodeq1d 15956 | . 2 ⊢ (𝑛 = 𝑁 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 − 𝑘) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − 𝑘)) |
| 6 | df-fallfac 16043 | . 2 ⊢ FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘)) | |
| 7 | prodex 15941 | . 2 ⊢ ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − 𝑘) ∈ V | |
| 8 | 2, 5, 6, 7 | ovmpo 7593 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 − 𝑘)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 − cmin 11492 ℕ0cn0 12526 ...cfz 13547 ∏cprod 15939 FallFac cfallfac 16040 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seq 14043 df-prod 15940 df-fallfac 16043 |
| This theorem is referenced by: fallfacval2 16047 fallfacval3 16048 fallfaccllem 16050 fallfacp1 16066 fallfacfwd 16072 0fallfac 16073 bcled 42179 bcle2d 42180 bcc0 44359 |
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