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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 11026 | . . . . . 6 ⊢ ℕ ∈ V | |
2 | prmnn 15388 | . . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ) | |
3 | 2 | ssriv 3607 | . . . . . 6 ⊢ ℙ ⊆ ℕ |
4 | 1, 3 | ssexi 4803 | . . . . 5 ⊢ ℙ ∈ V |
5 | 4 | rabex 4813 | . . . 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 4657 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴)) | |
9 | 8 | rabbidv 3189 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) |
10 | vmaval.1 | . . . . . . . 8 ⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} | |
11 | 9, 10 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = 𝑆) |
12 | 7, 11 | sylan9eqr 2678 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → 𝑠 = 𝑆) |
13 | 12 | fveq2d 6195 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (#‘𝑠) = (#‘𝑆)) |
14 | 13 | eqeq1d 2624 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ((#‘𝑠) = 1 ↔ (#‘𝑆) = 1)) |
15 | 12 | unieqd 4446 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ∪ 𝑠 = ∪ 𝑆) |
16 | 15 | fveq2d 6195 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (log‘∪ 𝑠) = (log‘∪ 𝑆)) |
17 | 14, 16 | ifbieq1d 4109 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → if((#‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
18 | 6, 17 | csbied 3560 | . 2 ⊢ (𝑥 = 𝐴 → ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
19 | df-vma 24824 | . 2 ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠), 0)) | |
20 | fvex 6201 | . . 3 ⊢ (log‘∪ 𝑆) ∈ V | |
21 | c0ex 10034 | . . 3 ⊢ 0 ∈ V | |
22 | 20, 21 | ifex 4156 | . 2 ⊢ if((#‘𝑆) = 1, (log‘∪ 𝑆), 0) ∈ V |
23 | 18, 19, 22 | fvmpt 6282 | 1 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((#‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 ⦋csb 3533 ifcif 4086 ∪ cuni 4436 class class class wbr 4653 ‘cfv 5888 0cc0 9936 1c1 9937 ℕcn 11020 #chash 13117 ∥ cdvds 14983 ℙcprime 15385 logclog 24301 Λcvma 24818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-prm 15386 df-vma 24824 |
This theorem is referenced by: isppw 24840 vmappw 24842 |
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