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Mirrors > Home > MPE Home > Th. List > vma1 | Structured version Visualization version GIF version |
Description: The von Mangoldt function at 1. (Contributed by Mario Carneiro, 9-Apr-2016.) |
Ref | Expression |
---|---|
vma1 | ⊢ (Λ‘1) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 10493 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ) | |
2 | prmuz2 15874 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ≥‘2)) | |
3 | 2 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ (ℤ≥‘2)) |
4 | eluz2b2 12175 | . . . . . . . . . 10 ⊢ (𝑝 ∈ (ℤ≥‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝)) | |
5 | 3, 4 | sylib 219 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (𝑝 ∈ ℕ ∧ 1 < 𝑝)) |
6 | 5 | simpld 495 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℕ) |
7 | 6 | nnred 11506 | . . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℝ) |
8 | nnnn0 11757 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
9 | 8 | adantl 482 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0) |
10 | 7, 9 | reexpcld 13382 | . . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ∈ ℝ) |
11 | 5 | simprd 496 | . . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 1 < 𝑝) |
12 | 6 | nncnd 11507 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℂ) |
13 | 12 | exp1d 13360 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (𝑝↑1) = 𝑝) |
14 | 6 | nnge1d 11538 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑝) |
15 | simpr 485 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
16 | nnuz 12135 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | syl6eleq 2893 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1)) |
18 | 7, 14, 17 | leexp2ad 13472 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (𝑝↑1) ≤ (𝑝↑𝑘)) |
19 | 13, 18 | eqbrtrrd 4990 | . . . . . . 7 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 𝑝 ≤ (𝑝↑𝑘)) |
20 | 1, 7, 10, 11, 19 | ltletrd 10652 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 1 < (𝑝↑𝑘)) |
21 | 1, 20 | ltned 10628 | . . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → 1 ≠ (𝑝↑𝑘)) |
22 | 21 | neneqd 2989 | . . . 4 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → ¬ 1 = (𝑝↑𝑘)) |
23 | 22 | nrexdv 3233 | . . 3 ⊢ (𝑝 ∈ ℙ → ¬ ∃𝑘 ∈ ℕ 1 = (𝑝↑𝑘)) |
24 | 23 | nrex 3232 | . 2 ⊢ ¬ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 1 = (𝑝↑𝑘) |
25 | 1nn 11502 | . . . 4 ⊢ 1 ∈ ℕ | |
26 | isppw2 25379 | . . . 4 ⊢ (1 ∈ ℕ → ((Λ‘1) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 1 = (𝑝↑𝑘))) | |
27 | 25, 26 | ax-mp 5 | . . 3 ⊢ ((Λ‘1) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 1 = (𝑝↑𝑘)) |
28 | 27 | necon1bbii 3033 | . 2 ⊢ (¬ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 1 = (𝑝↑𝑘) ↔ (Λ‘1) = 0) |
29 | 24, 28 | mpbi 231 | 1 ⊢ (Λ‘1) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∃wrex 3106 class class class wbr 4966 ‘cfv 6230 (class class class)co 7021 0cc0 10388 1c1 10389 < clt 10526 ≤ cle 10527 ℕcn 11491 2c2 11545 ℕ0cn0 11750 ℤ≥cuz 12098 ↑cexp 13284 ℙcprime 15849 Λcvma 25356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-inf2 8955 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 ax-addf 10467 ax-mulf 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-oadd 7962 df-er 8144 df-map 8263 df-pm 8264 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-fi 8726 df-sup 8757 df-inf 8758 df-oi 8825 df-dju 9181 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-q 12203 df-rp 12245 df-xneg 12362 df-xadd 12363 df-xmul 12364 df-ioo 12597 df-ioc 12598 df-ico 12599 df-icc 12600 df-fz 12748 df-fzo 12889 df-fl 13017 df-mod 13093 df-seq 13225 df-exp 13285 df-fac 13489 df-bc 13518 df-hash 13546 df-shft 14265 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 df-limsup 14667 df-clim 14684 df-rlim 14685 df-sum 14882 df-ef 15259 df-sin 15261 df-cos 15262 df-pi 15264 df-dvds 15446 df-gcd 15682 df-prm 15850 df-pc 16008 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-starv 16414 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-unif 16422 df-hom 16423 df-cco 16424 df-rest 16530 df-topn 16531 df-0g 16549 df-gsum 16550 df-topgen 16551 df-pt 16552 df-prds 16555 df-xrs 16609 df-qtop 16614 df-imas 16615 df-xps 16617 df-mre 16691 df-mrc 16692 df-acs 16694 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-submnd 17780 df-mulg 17987 df-cntz 18193 df-cmn 18640 df-psmet 20224 df-xmet 20225 df-met 20226 df-bl 20227 df-mopn 20228 df-fbas 20229 df-fg 20230 df-cnfld 20233 df-top 21191 df-topon 21208 df-topsp 21230 df-bases 21243 df-cld 21316 df-ntr 21317 df-cls 21318 df-nei 21395 df-lp 21433 df-perf 21434 df-cn 21524 df-cnp 21525 df-haus 21612 df-tx 21859 df-hmeo 22052 df-fil 22143 df-fm 22235 df-flim 22236 df-flf 22237 df-xms 22618 df-ms 22619 df-tms 22620 df-cncf 23174 df-limc 24152 df-dv 24153 df-log 24826 df-vma 25362 |
This theorem is referenced by: chp1 25431 |
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