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Theorem prmoval 17072
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 7440 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21prodeq1d 15957 . 2 (𝑛 = 𝑁 → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
3 df-prmo 17071 . 2 #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
4 prodex 15942 . 2 𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ V
52, 3, 4fvmpt 7015 1 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  ifcif 4524  cfv 6560  (class class class)co 7432  1c1 11157  0cn0 12528  ...cfz 13548  cprod 15940  cprime 16709  #pcprmo 17070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fun 6562  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seq 14044  df-prod 15941  df-prmo 17071
This theorem is referenced by:  prmocl  17073  prmo0  17075  prmo1  17076  prmop1  17077  prmdvdsprmo  17081  prmolefac  17085  prmodvdslcmf  17086  prmgapprmo  17101
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