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Mirrors > Home > MPE Home > Th. List > musumsum | Structured version Visualization version GIF version |
Description: Evaluate a collapsing sum over the Möbius function. (Contributed by Mario Carneiro, 4-May-2016.) |
Ref | Expression |
---|---|
musumsum.1 | ⊢ (𝑚 = 1 → 𝐵 = 𝐶) |
musumsum.2 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
musumsum.3 | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
musumsum.4 | ⊢ (𝜑 → 1 ∈ 𝐴) |
musumsum.5 | ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
musumsum | ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | musumsum.3 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
2 | 1 | sselda 3915 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ ℕ) |
3 | musum 25776 | . . . . . 6 ⊢ (𝑚 ∈ ℕ → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) = if(𝑚 = 1, 1, 0)) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) = if(𝑚 = 1, 1, 0)) |
5 | 4 | oveq1d 7150 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = (if(𝑚 = 1, 1, 0) · 𝐵)) |
6 | fzfid 13336 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (1...𝑚) ∈ Fin) | |
7 | dvdsssfz1 15660 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ⊆ (1...𝑚)) | |
8 | 2, 7 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ⊆ (1...𝑚)) |
9 | 6, 8 | ssfid 8725 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ∈ Fin) |
10 | musumsum.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
11 | elrabi 3623 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → 𝑘 ∈ ℕ) | |
12 | mucl 25726 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ) | |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℤ) |
14 | 13 | zcnd 12076 | . . . . . 6 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℂ) |
15 | 14 | adantl 485 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚}) → (μ‘𝑘) ∈ ℂ) |
16 | 9, 10, 15 | fsummulc1 15132 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵)) |
17 | ovif 7230 | . . . . 5 ⊢ (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) | |
18 | velsn 4541 | . . . . . . . . 9 ⊢ (𝑚 ∈ {1} ↔ 𝑚 = 1) | |
19 | 18 | bicomi 227 | . . . . . . . 8 ⊢ (𝑚 = 1 ↔ 𝑚 ∈ {1}) |
20 | 19 | a1i 11 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (𝑚 = 1 ↔ 𝑚 ∈ {1})) |
21 | mulid2 10629 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
22 | mul02 10807 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (0 · 𝐵) = 0) | |
23 | 20, 21, 22 | ifbieq12d 4452 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) = if(𝑚 ∈ {1}, 𝐵, 0)) |
24 | 10, 23 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) = if(𝑚 ∈ {1}, 𝐵, 0)) |
25 | 17, 24 | syl5eq 2845 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
26 | 5, 16, 25 | 3eqtr3d 2841 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
27 | 26 | sumeq2dv 15052 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = Σ𝑚 ∈ 𝐴 if(𝑚 ∈ {1}, 𝐵, 0)) |
28 | musumsum.4 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝐴) | |
29 | 28 | snssd 4702 | . . 3 ⊢ (𝜑 → {1} ⊆ 𝐴) |
30 | 29 | sselda 3915 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝑚 ∈ 𝐴) |
31 | 30, 10 | syldan 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝐵 ∈ ℂ) |
32 | 31 | ralrimiva 3149 | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ {1}𝐵 ∈ ℂ) |
33 | musumsum.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
34 | 33 | olcd 871 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ (ℤ≥‘1) ∨ 𝐴 ∈ Fin)) |
35 | sumss2 15075 | . . 3 ⊢ ((({1} ⊆ 𝐴 ∧ ∀𝑚 ∈ {1}𝐵 ∈ ℂ) ∧ (𝐴 ⊆ (ℤ≥‘1) ∨ 𝐴 ∈ Fin)) → Σ𝑚 ∈ {1}𝐵 = Σ𝑚 ∈ 𝐴 if(𝑚 ∈ {1}, 𝐵, 0)) | |
36 | 29, 32, 34, 35 | syl21anc 836 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ {1}𝐵 = Σ𝑚 ∈ 𝐴 if(𝑚 ∈ {1}, 𝐵, 0)) |
37 | musumsum.1 | . . . . 5 ⊢ (𝑚 = 1 → 𝐵 = 𝐶) | |
38 | 37 | eleq1d 2874 | . . . 4 ⊢ (𝑚 = 1 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
39 | 10 | ralrimiva 3149 | . . . 4 ⊢ (𝜑 → ∀𝑚 ∈ 𝐴 𝐵 ∈ ℂ) |
40 | 38, 39, 28 | rspcdva 3573 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
41 | 37 | sumsn 15093 | . . 3 ⊢ ((1 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑚 ∈ {1}𝐵 = 𝐶) |
42 | 28, 40, 41 | syl2anc 587 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ {1}𝐵 = 𝐶) |
43 | 27, 36, 42 | 3eqtr2d 2839 | 1 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ⊆ wss 3881 ifcif 4425 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 0cc0 10526 1c1 10527 · cmul 10531 ℕcn 11625 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12885 Σcsu 15034 ∥ cdvds 15599 μcmu 25680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-dvds 15600 df-gcd 15834 df-prm 16006 df-pc 16164 df-mu 25686 |
This theorem is referenced by: dchrmusum2 26078 dchrvmasum2lem 26080 mudivsum 26114 mulogsum 26116 mulog2sumlem2 26119 |
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