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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 7404 | . . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | 1 | prodeq1d 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 | |
5 | 2, 3, 4 | fvmpt 6987 | 1 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏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 ℕ0cn0 12459 ...cfz 13471 ∏cprod 15836 ℙcprime 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 |
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