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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 7440 | . . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
| 2 | 1 | prodeq1d 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 | |
| 5 | 2, 3, 4 | fvmpt 7015 | 1 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏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 ℕ0cn0 12528 ...cfz 13548 ∏cprod 15940 ℙcprime 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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