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Mirrors > Home > MPE Home > Th. List > vmappw | Structured version Visualization version GIF version |
Description: Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.) |
Ref | Expression |
---|---|
vmappw | ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃↑𝐾)) = (log‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmnn 16668 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
2 | nnnn0 12523 | . . . 4 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℕ0) | |
3 | nnexpcl 14086 | . . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝐾 ∈ ℕ0) → (𝑃↑𝐾) ∈ ℕ) | |
4 | 1, 2, 3 | syl2an 594 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (𝑃↑𝐾) ∈ ℕ) |
5 | eqid 2726 | . . . 4 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)} | |
6 | 5 | vmaval 27136 | . . 3 ⊢ ((𝑃↑𝐾) ∈ ℕ → (Λ‘(𝑃↑𝐾)) = if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}), 0)) |
7 | 4, 6 | syl 17 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃↑𝐾)) = if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}), 0)) |
8 | df-rab 3421 | . . . . . 6 ⊢ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)} = {𝑝 ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑃↑𝐾))} | |
9 | prmdvdsexpb 16710 | . . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (𝑝 ∥ (𝑃↑𝐾) ↔ 𝑝 = 𝑃)) | |
10 | 9 | biimpd 228 | . . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℙ ∧ 𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (𝑝 ∥ (𝑃↑𝐾) → 𝑝 = 𝑃)) |
11 | 10 | 3coml 1124 | . . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝑃↑𝐾) → 𝑝 = 𝑃)) |
12 | 11 | 3expa 1115 | . . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) ∧ 𝑝 ∈ ℙ) → (𝑝 ∥ (𝑃↑𝐾) → 𝑝 = 𝑃)) |
13 | 12 | expimpd 452 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑃↑𝐾)) → 𝑝 = 𝑃)) |
14 | simpl 481 | . . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → 𝑃 ∈ ℙ) | |
15 | prmz 16669 | . . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
16 | iddvdsexp 16275 | . . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℤ ∧ 𝐾 ∈ ℕ) → 𝑃 ∥ (𝑃↑𝐾)) | |
17 | 15, 16 | sylan 578 | . . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → 𝑃 ∥ (𝑃↑𝐾)) |
18 | 14, 17 | jca 510 | . . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝑃↑𝐾))) |
19 | eleq1 2814 | . . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ ℙ ↔ 𝑃 ∈ ℙ)) | |
20 | breq1 5147 | . . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → (𝑝 ∥ (𝑃↑𝐾) ↔ 𝑃 ∥ (𝑃↑𝐾))) | |
21 | 19, 20 | anbi12d 630 | . . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑃↑𝐾)) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝑃↑𝐾)))) |
22 | 18, 21 | syl5ibrcom 246 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (𝑝 = 𝑃 → (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑃↑𝐾)))) |
23 | 13, 22 | impbid 211 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑃↑𝐾)) ↔ 𝑝 = 𝑃)) |
24 | velsn 4640 | . . . . . . . 8 ⊢ (𝑝 ∈ {𝑃} ↔ 𝑝 = 𝑃) | |
25 | 23, 24 | bitr4di 288 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑃↑𝐾)) ↔ 𝑝 ∈ {𝑃})) |
26 | 25 | eqabcdv 2861 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → {𝑝 ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ (𝑃↑𝐾))} = {𝑃}) |
27 | 8, 26 | eqtrid 2778 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)} = {𝑃}) |
28 | 27 | fveq2d 6895 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}) = (♯‘{𝑃})) |
29 | hashsng 14379 | . . . . 5 ⊢ (𝑃 ∈ ℙ → (♯‘{𝑃}) = 1) | |
30 | 29 | adantr 479 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (♯‘{𝑃}) = 1) |
31 | 28, 30 | eqtrd 2766 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}) = 1) |
32 | 31 | iftrued 4532 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → if((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}) = 1, (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}), 0) = (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)})) |
33 | 27 | unieqd 4919 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)} = ∪ {𝑃}) |
34 | unisng 4926 | . . . . 5 ⊢ (𝑃 ∈ ℙ → ∪ {𝑃} = 𝑃) | |
35 | 34 | adantr 479 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → ∪ {𝑃} = 𝑃) |
36 | 33, 35 | eqtrd 2766 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → ∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)} = 𝑃) |
37 | 36 | fveq2d 6895 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (log‘∪ {𝑝 ∈ ℙ ∣ 𝑝 ∥ (𝑃↑𝐾)}) = (log‘𝑃)) |
38 | 7, 32, 37 | 3eqtrd 2770 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (Λ‘(𝑃↑𝐾)) = (log‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 {cab 2703 {crab 3420 ifcif 4524 {csn 4624 ∪ cuni 4906 class class class wbr 5144 ‘cfv 6544 (class class class)co 7414 0cc0 11147 1c1 11148 ℕcn 12256 ℕ0cn0 12516 ℤcz 12602 ↑cexp 14073 ♯chash 14340 ∥ cdvds 16249 ℙcprime 16665 logclog 26576 Λcvma 27115 |
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 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 |
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-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9476 df-inf 9477 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-n0 12517 df-z 12603 df-uz 12867 df-rp 13021 df-fz 13531 df-fl 13804 df-mod 13882 df-seq 14014 df-exp 14074 df-hash 14341 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-dvds 16250 df-gcd 16488 df-prm 16666 df-vma 27121 |
This theorem is referenced by: vmaprm 27140 vmacl 27141 efvmacl 27143 vmalelog 27229 vmasum 27240 chpval2 27242 rplogsumlem2 27509 rpvmasumlem 27511 |
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