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Mirrors > Home > MPE Home > Th. List > mule1 | Structured version Visualization version GIF version |
Description: The Möbius function takes on values in magnitude at most 1. (Together with mucl 25219, this implies that it takes a value in {-1, 0, 1} for every positive integer.) (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
mule1 | ⊢ (𝐴 ∈ ℕ → (abs‘(μ‘𝐴)) ≤ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muval 25210 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) | |
2 | iftrue 4283 | . . . . 5 ⊢ (∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 0) | |
3 | 1, 2 | sylan9eq 2853 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (μ‘𝐴) = 0) |
4 | 3 | fveq2d 6415 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) = (abs‘0)) |
5 | abs0 14366 | . . . 4 ⊢ (abs‘0) = 0 | |
6 | 0le1 10843 | . . . 4 ⊢ 0 ≤ 1 | |
7 | 5, 6 | eqbrtri 4864 | . . 3 ⊢ (abs‘0) ≤ 1 |
8 | 4, 7 | syl6eqbr 4882 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) ≤ 1) |
9 | iffalse 4286 | . . . . . 6 ⊢ (¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) | |
10 | 1, 9 | sylan9eq 2853 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) |
11 | 10 | fveq2d 6415 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) = (abs‘(-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
12 | neg1cn 11434 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
13 | prmdvdsfi 25185 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin) | |
14 | hashcl 13397 | . . . . . . . 8 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ0) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ0) |
16 | absexp 14385 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ0) → (abs‘(-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = ((abs‘-1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) | |
17 | 12, 15, 16 | sylancr 582 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (abs‘(-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = ((abs‘-1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) |
18 | ax-1cn 10282 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
19 | 18 | absnegi 14480 | . . . . . . . . 9 ⊢ (abs‘-1) = (abs‘1) |
20 | abs1 14378 | . . . . . . . . 9 ⊢ (abs‘1) = 1 | |
21 | 19, 20 | eqtri 2821 | . . . . . . . 8 ⊢ (abs‘-1) = 1 |
22 | 21 | oveq1i 6888 | . . . . . . 7 ⊢ ((abs‘-1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) = (1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) |
23 | 15 | nn0zd 11770 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℤ) |
24 | 1exp 13143 | . . . . . . . 8 ⊢ ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℤ → (1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) = 1) | |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → (1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) = 1) |
26 | 22, 25 | syl5eq 2845 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((abs‘-1)↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) = 1) |
27 | 17, 26 | eqtrd 2833 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (abs‘(-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 1) |
28 | 27 | adantr 473 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 1) |
29 | 11, 28 | eqtrd 2833 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) = 1) |
30 | 1le1 10947 | . . 3 ⊢ 1 ≤ 1 | |
31 | 29, 30 | syl6eqbr 4882 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) ≤ 1) |
32 | 8, 31 | pm2.61dan 848 | 1 ⊢ (𝐴 ∈ ℕ → (abs‘(μ‘𝐴)) ≤ 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 {crab 3093 ifcif 4277 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 Fincfn 8195 ℂcc 10222 0cc0 10224 1c1 10225 ≤ cle 10364 -cneg 10557 ℕcn 11312 2c2 11368 ℕ0cn0 11580 ℤcz 11666 ↑cexp 13114 ♯chash 13370 abscabs 14315 ∥ cdvds 15319 ℙcprime 15719 μcmu 25173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-fz 12581 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-dvds 15320 df-prm 15720 df-mu 25179 |
This theorem is referenced by: dchrmusum2 25535 dchrvmasumlem3 25540 mudivsum 25571 mulogsumlem 25572 mulog2sumlem2 25576 selberglem2 25587 |
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