Theorem risefacval 16040

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 7437	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑘) = (𝐴 + 𝑘))
2	1	prodeq2sdv 15955	. 2 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘) = ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 + 𝑘))
3		oveq1 7437	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 − 1) = (𝑁 − 1))
4	3	oveq2d 7446	. . 3 ⊢ (𝑛 = 𝑁 → (0...(𝑛 − 1)) = (0...(𝑁 − 1)))
5	4	prodeq1d 15952	. 2 ⊢ (𝑛 = 𝑁 → ∏𝑘 ∈ (0...(𝑛 − 1))(𝐴 + 𝑘) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))
6		df-risefac 16038	. 2 ⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘))
7		prodex 15937	. 2 ⊢ ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘) ∈ V
8	2, 5, 6, 7	ovmpo 7592	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴 RiseFac 𝑁) = ∏𝑘 ∈ (0...(𝑁 − 1))(𝐴 + 𝑘))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  (class class class)co 7430  ℂcc 11150  0cc0 11152  1c1 11153   + caddc 11155   − cmin 11489  ℕ₀cn0 12523  ...cfz 13543  ∏cprod 15935   RiseFac crisefac 16037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-iota 6515  df-fun 6564  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seq 14039  df-prod 15936  df-risefac 16038

This theorem is referenced by:  risefacval2  16042  risefaccllem  16045  risefac0  16059  risefacp1  16061