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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 | df-prmo 15574 | . . 3 ⊢ #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → #p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))) |
3 | oveq2 6557 | . . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
4 | 3 | prodeq1d 14490 | . . 3 ⊢ (𝑛 = 𝑁 → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1)) |
5 | 4 | adantl 481 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑛 = 𝑁) → ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1)) |
6 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
7 | prodex 14476 | . . 3 ⊢ ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ V | |
8 | 7 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1) ∈ V) |
9 | 2, 5, 6, 8 | fvmptd 6197 | 1 ⊢ (𝑁 ∈ ℕ0 → (#p‘𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 1c1 9816 ℕ0cn0 11169 ...cfz 12197 ∏cprod 14474 ℙcprime 15223 #pcprmo 15573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-prod 14475 df-prmo 15574 |
This theorem is referenced by: prmocl 15576 prmo0 15578 prmo1 15579 prmop1 15580 prmdvdsprmo 15584 prmolefac 15588 prmodvdslcmf 15589 prmgapprmo 15604 |
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