Theorem risefacval 15921

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.

Step	Hyp	Ref	Expression
1		oveq1 7359	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑘) = (𝐴 + 𝑘))
2	1	prodeq2sdv 15836	. 2 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘) = ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 + 𝑘))
3		oveq1 7359	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
4	3	oveq2d 7368	. . 3 ⊢ (𝑛 = 𝑁 → (0...(𝑛 − 1)) = (0...(𝑁 − 1)))
5	4	prodeq1d 15833	. 2 ⊢ (𝑛 = 𝑁 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 + 𝑘) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))
6		df-risefac 15919	. 2 ⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
7		prodex 15818	. 2 ⊢ ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘) ∈ V
8	2, 5, 6, 7	ovmpo 7512	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  (class class class)co 7352  ℂcc 11010  0cc0 11012  1c1 11013   + caddc 11015   − cmin 11350  ℕ₀cn0 12387  ...cfz 13413  ∏cprod 15816   RiseFac crisefac 15918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-iota 6443  df-fun 6489  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-seq 13915  df-prod 15817  df-risefac 15919

This theorem is referenced by:  risefacval2  15923  risefaccllem  15926  risefac0  15940  risefacp1  15942