Theorem risefacval 16024

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 7412	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑘) = (𝐴 + 𝑘))
2	1	prodeq2sdv 15939	. 2 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘) = ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 + 𝑘))
3		oveq1 7412	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
4	3	oveq2d 7421	. . 3 ⊢ (𝑛 = 𝑁 → (0...(𝑛 − 1)) = (0...(𝑁 − 1)))
5	4	prodeq1d 15936	. 2 ⊢ (𝑛 = 𝑁 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 + 𝑘) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))
6		df-risefac 16022	. 2 ⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
7		prodex 15921	. 2 ⊢ ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘) ∈ V
8	2, 5, 6, 7	ovmpo 7567	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  (class class class)co 7405  ℂcc 11127  0cc0 11129  1c1 11130   + caddc 11132   − cmin 11466  ℕ₀cn0 12501  ...cfz 13524  ∏cprod 15919   RiseFac crisefac 16021

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-iota 6484  df-fun 6533  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seq 14020  df-prod 15920  df-risefac 16022

This theorem is referenced by:  risefacval2  16026  risefaccllem  16029  risefac0  16043  risefacp1  16045