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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 | prmex 16678 | . . . . 5 ⊢ ℙ ∈ V | |
2 | 1 | rabex 5339 | . . . 4 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} ∈ V) |
4 | id 22 | . . . . . 6 ⊢ (𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} → 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) | |
5 | breq2 5157 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴)) | |
6 | 5 | rabbidv 3427 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) |
7 | vmaval.1 | . . . . . . 7 ⊢ 𝑆 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} | |
8 | 6, 7 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = 𝑆) |
9 | 4, 8 | sylan9eqr 2788 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → 𝑠 = 𝑆) |
10 | 9 | fveqeq2d 6909 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ((♯‘𝑠) = 1 ↔ (♯‘𝑆) = 1)) |
11 | 9 | unieqd 4926 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → ∪ 𝑠 = ∪ 𝑆) |
12 | 11 | fveq2d 6905 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → (log‘∪ 𝑠) = (log‘∪ 𝑆)) |
13 | 10, 12 | ifbieq1d 4557 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑠 = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) → if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
14 | 3, 13 | csbied 3930 | . 2 ⊢ (𝑥 = 𝐴 → ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
15 | df-vma 27126 | . 2 ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((♯‘𝑠) = 1, (log‘∪ 𝑠), 0)) | |
16 | fvex 6914 | . . 3 ⊢ (log‘∪ 𝑆) ∈ V | |
17 | c0ex 11258 | . . 3 ⊢ 0 ∈ V | |
18 | 16, 17 | ifex 4583 | . 2 ⊢ if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0) ∈ V |
19 | 14, 15, 18 | fvmpt 7009 | 1 ⊢ (𝐴 ∈ ℕ → (Λ‘𝐴) = if((♯‘𝑆) = 1, (log‘∪ 𝑆), 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 ⦋csb 3892 ifcif 4533 ∪ cuni 4913 class class class wbr 5153 ‘cfv 6554 0cc0 11158 1c1 11159 ℕcn 12264 ♯chash 14347 ∥ cdvds 16256 ℙcprime 16672 logclog 26581 Λcvma 27120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-mulcl 11220 ax-i2m1 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-nn 12265 df-prm 16673 df-vma 27126 |
This theorem is referenced by: isppw 27142 vmappw 27144 |
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