![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > prmoval | Structured version Visualization version GIF version |
Description: Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.) |
Ref | Expression |
---|---|
prmoval | ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7420 | . . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | 1 | prodeq1d 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 | |
5 | 2, 3, 4 | fvmpt 6998 | 1 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏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 ℕ0cn0 12477 ...cfz 13489 ∏cprod 15854 ℙcprime 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 |
Copyright terms: Public domain | W3C validator |