MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prmoval Structured version   Visualization version   GIF version

Theorem prmoval 16953
Description: Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.)
Assertion
Ref Expression
prmoval (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
Distinct variable group:   𝑘,𝑁

Proof of Theorem prmoval
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7404 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21prodeq1d 15852 . 2 (𝑛 = 𝑁 → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
3 df-prmo 16952 . 2 #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
4 prodex 15838 . 2 𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ V
52, 3, 4fvmpt 6987 1 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  ifcif 4524  cfv 6535  (class class class)co 7396  1c1 11098  0cn0 12459  ...cfz 13471  cprod 15836  cprime 16595  #pcprmo 16951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-iota 6487  df-fun 6537  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-seq 13954  df-prod 15837  df-prmo 16952
This theorem is referenced by:  prmocl  16954  prmo0  16956  prmo1  16957  prmop1  16958  prmdvdsprmo  16962  prmolefac  16966  prmodvdslcmf  16967  prmgapprmo  16982
  Copyright terms: Public domain W3C validator