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Theorem risefacval 16044
Description: The value of the rising factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
Assertion
Ref Expression
risefacval ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))
Distinct variable groups:   𝐴,𝑘   𝑘,𝑁

Proof of Theorem risefacval
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . 3 (𝑥 = 𝐴 → (𝑥 + 𝑘) = (𝐴 + 𝑘))
21prodeq2sdv 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))
43oveq2d 7447 . . 3 (𝑛 = 𝑁 → (0...(𝑛 − 1)) = (0...(𝑁 − 1)))
54prodeq1d 15956 . 2 (𝑛 = 𝑁 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 + 𝑘) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))
6 df-risefac 16042 . 2 RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
7 prodex 15941 . 2 𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘) ∈ V
82, 5, 6, 7ovmpo 7593 1 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 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  cc 11153  0cc0 11155  1c1 11156   + caddc 11158  cmin 11492  0cn0 12526  ...cfz 13547  cprod 15939   RiseFac crisefac 16041
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-risefac 16042
This theorem is referenced by:  risefacval2  16046  risefaccllem  16049  risefac0  16063  risefacp1  16065
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