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Theorem prmoval 16971
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 7420 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21prodeq1d 15870 . 2 (𝑛 = 𝑁 → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
3 df-prmo 16970 . 2 #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
4 prodex 15856 . 2 𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ V
52, 3, 4fvmpt 6998 1 (𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  ifcif 4528  cfv 6543  (class class class)co 7412  1c1 11114  0cn0 12477  ...cfz 13489  cprod 15854  cprime 16613  #pcprmo 16969
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-seq 13972  df-prod 15855  df-prmo 16970
This theorem is referenced by:  prmocl  16972  prmo0  16974  prmo1  16975  prmop1  16976  prmdvdsprmo  16980  prmolefac  16984  prmodvdslcmf  16985  prmgapprmo  17000
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