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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 24867, 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 24858 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) | |
2 | iftrue 4092 | . . . . 5 ⊢ (∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 0) | |
3 | 1, 2 | sylan9eq 2676 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (μ‘𝐴) = 0) |
4 | 3 | fveq2d 6195 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) = (abs‘0)) |
5 | abs0 14025 | . . . 4 ⊢ (abs‘0) = 0 | |
6 | 0le1 10551 | . . . 4 ⊢ 0 ≤ 1 | |
7 | 5, 6 | eqbrtri 4674 | . . 3 ⊢ (abs‘0) ≤ 1 |
8 | 4, 7 | syl6eqbr 4692 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) ≤ 1) |
9 | iffalse 4095 | . . . . . 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 2676 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (μ‘𝐴) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) |
11 | 10 | fveq2d 6195 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) = (abs‘(-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
12 | neg1cn 11124 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
13 | prmdvdsfi 24833 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin) | |
14 | hashcl 13147 | . . . . . . . 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 14044 | . . . . . . 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 695 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (abs‘(-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = ((abs‘-1)↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) |
18 | ax-1cn 9994 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
19 | 18 | absnegi 14139 | . . . . . . . . 9 ⊢ (abs‘-1) = (abs‘1) |
20 | abs1 14037 | . . . . . . . . 9 ⊢ (abs‘1) = 1 | |
21 | 19, 20 | eqtri 2644 | . . . . . . . 8 ⊢ (abs‘-1) = 1 |
22 | 21 | oveq1i 6660 | . . . . . . 7 ⊢ ((abs‘-1)↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) = (1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) |
23 | 15 | nn0zd 11480 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℤ) |
24 | 1exp 12889 | . . . . . . . 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 2668 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((abs‘-1)↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) = 1) |
27 | 17, 26 | eqtrd 2656 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (abs‘(-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 1) |
28 | 27 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 1) |
29 | 11, 28 | eqtrd 2656 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) = 1) |
30 | 1le1 10655 | . . 3 ⊢ 1 ≤ 1 | |
31 | 29, 30 | syl6eqbr 4692 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) → (abs‘(μ‘𝐴)) ≤ 1) |
32 | 8, 31 | pm2.61dan 832 | 1 ⊢ (𝐴 ∈ ℕ → (abs‘(μ‘𝐴)) ≤ 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 {crab 2916 ifcif 4086 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 0cc0 9936 1c1 9937 ≤ cle 10075 -cneg 10267 ℕcn 11020 2c2 11070 ℕ0cn0 11292 ℤcz 11377 ↑cexp 12860 #chash 13117 abscabs 13974 ∥ cdvds 14983 ℙcprime 15385 μcmu 24821 |
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-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 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-int 4476 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-prm 15386 df-mu 24827 |
This theorem is referenced by: dchrmusum2 25183 dchrvmasumlem3 25188 mudivsum 25219 mulogsumlem 25220 mulog2sumlem2 25224 selberglem2 25235 |
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