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Mirrors > Home > MPE Home > Th. List > vmaval | Structured version Visualization version GIF version |
Description: Value of the von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.) |
Ref | Expression |
---|---|
vmaval.1 | ⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} |
Ref | Expression |
---|---|
vmaval | ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 10903 | . . . . . 6 ⊢ ℕ ∈ V | |
2 | prmnn 15226 | . . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ) | |
3 | 2 | ssriv 3572 | . . . . . 6 ⊢ ℙ ⊆ ℕ |
4 | 1, 3 | ssexi 4731 | . . . . 5 ⊢ ℙ ∈ V |
5 | 4 | rabex 4740 | . . . 4 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} ∈ V |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} ∈ V) |
7 | id 22 | . . . . . . 7 ⊢ (𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} → 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) | |
8 | breq2 4587 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴)) | |
9 | 8 | rabbidv 3164 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) |
10 | vmaval.1 | . . . . . . . 8 ⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} | |
11 | 9, 10 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = 𝑆) |
12 | 7, 11 | sylan9eqr 2666 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → 𝑠 = 𝑆) |
13 | 12 | fveq2d 6107 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (#‘𝑠) = (#‘𝑆)) |
14 | 13 | eqeq1d 2612 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ((#‘𝑠) = 1 ↔ (#‘𝑆) = 1)) |
15 | 12 | unieqd 4382 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ∪ 𝑠 = ∪ 𝑆) |
16 | 15 | fveq2d 6107 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (log‘∪ 𝑠) = (log‘∪ 𝑆)) |
17 | 14, 16 | ifbieq1d 4059 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → if((#‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
18 | 6, 17 | csbied 3526 | . 2 ⊢ (𝑥 = 𝐴 → ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
19 | df-vma 24624 | . 2 ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0)) | |
20 | fvex 6113 | . . 3 ⊢ (log‘∪ 𝑆) ∈ V | |
21 | c0ex 9913 | . . 3 ⊢ 0 ∈ V | |
22 | 20, 21 | ifex 4106 | . 2 ⊢ if((#‘𝑆) = 1, (log‘∪ 𝑆), 0) ∈ V |
23 | 18, 19, 22 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⦋csb 3499 ifcif 4036 ∪ cuni 4372 class class class wbr 4583 ‘cfv 5804 0cc0 9815 1c1 9816 ℕcn 10897 #chash 12979 ∥ cdvds 14821 ℙcprime 15223 logclog 24105 Λcvma 24618 |
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-8 1979 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-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-prm 15224 df-vma 24624 |
This theorem is referenced by: isppw 24640 vmappw 24642 |
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