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Theorem fallfacval 16045
Description: The value of the falling factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
Assertion
Ref Expression
fallfacval ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴𝑘))
Distinct variable groups:   𝐴,𝑘   𝑘,𝑁

Proof of Theorem fallfacval
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . 3 (𝑥 = 𝐴 → (𝑥𝑘) = (𝐴𝑘))
21prodeq2sdv 15959 . 2 (𝑥 = 𝐴 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘) = ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴𝑘))
3 oveq1 7438 . . . 4 (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
43oveq2d 7447 . . 3 (𝑛 = 𝑁 → (0...(𝑛 − 1)) = (0...(𝑁 − 1)))
54prodeq1d 15956 . 2 (𝑛 = 𝑁 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴𝑘) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴𝑘))
6 df-fallfac 16043 . 2 FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥𝑘))
7 prodex 15941 . 2 𝑘 ∈ (0...(𝑁 − 1))(𝐴𝑘) ∈ V
82, 5, 6, 7ovmpo 7593 1 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 FallFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴𝑘))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  (class class class)co 7431  cc 11153  0cc0 11155  1c1 11156  cmin 11492  0cn0 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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