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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 3921 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ ℕ) |
3 | musum 26340 | . . . . . 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 7290 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = (if(𝑚 = 1, 1, 0) · 𝐵)) |
6 | fzfid 13693 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (1...𝑚) ∈ Fin) | |
7 | dvdsssfz1 16027 | . . . . . . 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 9042 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ∈ Fin) |
10 | musumsum.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
11 | elrabi 3618 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → 𝑘 ∈ ℕ) | |
12 | mucl 26290 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ) | |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℤ) |
14 | 13 | zcnd 12427 | . . . . . 6 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℂ) |
15 | 14 | adantl 482 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚}) → (μ‘𝑘) ∈ ℂ) |
16 | 9, 10, 15 | fsummulc1 15497 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵)) |
17 | ovif 7372 | . . . . 5 ⊢ (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) | |
18 | velsn 4577 | . . . . . . . . 9 ⊢ (𝑚 ∈ {1} ↔ 𝑚 = 1) | |
19 | 18 | bicomi 223 | . . . . . . . 8 ⊢ (𝑚 = 1 ↔ 𝑚 ∈ {1}) |
20 | 19 | a1i 11 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (𝑚 = 1 ↔ 𝑚 ∈ {1})) |
21 | mulid2 10974 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
22 | mul02 11153 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (0 · 𝐵) = 0) | |
23 | 20, 21, 22 | ifbieq12d 4487 | . . . . . 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 | eqtrid 2790 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
26 | 5, 16, 25 | 3eqtr3d 2786 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
27 | 26 | sumeq2dv 15415 | . 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 4742 | . . 3 ⊢ (𝜑 → {1} ⊆ 𝐴) |
30 | 29 | sselda 3921 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝑚 ∈ 𝐴) |
31 | 30, 10 | syldan 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝐵 ∈ ℂ) |
32 | 31 | ralrimiva 3103 | . . 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 15438 | . . 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 835 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ {1}𝐵 = Σ𝑚 ∈ 𝐴 if(𝑚 ∈ {1}, 𝐵, 0)) |
37 | musumsum.1 | . . . . 5 ⊢ (𝑚 = 1 → 𝐵 = 𝐶) | |
38 | 37 | eleq1d 2823 | . . . 4 ⊢ (𝑚 = 1 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
39 | 10 | ralrimiva 3103 | . . . 4 ⊢ (𝜑 → ∀𝑚 ∈ 𝐴 𝐵 ∈ ℂ) |
40 | 38, 39, 28 | rspcdva 3562 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
41 | 37 | sumsn 15458 | . . 3 ⊢ ((1 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑚 ∈ {1}𝐵 = 𝐶) |
42 | 28, 40, 41 | syl2anc 584 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ {1}𝐵 = 𝐶) |
43 | 27, 36, 42 | 3eqtr2d 2784 | 1 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 ⊆ wss 3887 ifcif 4459 {csn 4561 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 0cc0 10871 1c1 10872 · cmul 10876 ℕcn 11973 ℤcz 12319 ℤ≥cuz 12582 ...cfz 13239 Σcsu 15397 ∥ cdvds 15963 μcmu 26244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-dvds 15964 df-gcd 16202 df-prm 16377 df-pc 16538 df-mu 26250 |
This theorem is referenced by: dchrmusum2 26642 dchrvmasum2lem 26644 mudivsum 26678 mulogsum 26680 mulog2sumlem2 26683 |
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