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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 3568 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ ℕ) |
3 | musum 24717 | . . . . . 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 6564 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = (if(𝑚 = 1, 1, 0) · 𝐵)) |
6 | fzfid 12634 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (1...𝑚) ∈ Fin) | |
7 | dvdsssfz1 14878 | . . . . . . 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 | ssfi 8065 | . . . . . 6 ⊢ (((1...𝑚) ∈ Fin ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ⊆ (1...𝑚)) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ∈ Fin) | |
10 | 6, 8, 9 | syl2anc 691 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ∈ Fin) |
11 | musumsum.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
12 | elrabi 3328 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → 𝑘 ∈ ℕ) | |
13 | mucl 24667 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℤ) |
15 | 14 | zcnd 11359 | . . . . . 6 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℂ) |
16 | 15 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚}) → (μ‘𝑘) ∈ ℂ) |
17 | 10, 11, 16 | fsummulc1 14359 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵)) |
18 | ovif 6635 | . . . . 5 ⊢ (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) | |
19 | velsn 4141 | . . . . . . . . 9 ⊢ (𝑚 ∈ {1} ↔ 𝑚 = 1) | |
20 | 19 | bicomi 213 | . . . . . . . 8 ⊢ (𝑚 = 1 ↔ 𝑚 ∈ {1}) |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (𝑚 = 1 ↔ 𝑚 ∈ {1})) |
22 | mulid2 9917 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
23 | mul02 10093 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (0 · 𝐵) = 0) | |
24 | 21, 22, 23 | ifbieq12d 4063 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) = if(𝑚 ∈ {1}, 𝐵, 0)) |
25 | 11, 24 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) = if(𝑚 ∈ {1}, 𝐵, 0)) |
26 | 18, 25 | syl5eq 2656 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
27 | 5, 17, 26 | 3eqtr3d 2652 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
28 | 27 | sumeq2dv 14281 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = Σ𝑚 ∈ 𝐴 if(𝑚 ∈ {1}, 𝐵, 0)) |
29 | musumsum.4 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝐴) | |
30 | 29 | snssd 4281 | . . 3 ⊢ (𝜑 → {1} ⊆ 𝐴) |
31 | 30 | sselda 3568 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝑚 ∈ 𝐴) |
32 | 31, 11 | syldan 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝐵 ∈ ℂ) |
33 | 32 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ {1}𝐵 ∈ ℂ) |
34 | musumsum.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
35 | 34 | olcd 407 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ (ℤ≥‘1) ∨ 𝐴 ∈ Fin)) |
36 | sumss2 14304 | . . 3 ⊢ ((({1} ⊆ 𝐴 ∧ ∀𝑚 ∈ {1}𝐵 ∈ ℂ) ∧ (𝐴 ⊆ (ℤ≥‘1) ∨ 𝐴 ∈ Fin)) → Σ𝑚 ∈ {1}𝐵 = Σ𝑚 ∈ 𝐴 if(𝑚 ∈ {1}, 𝐵, 0)) | |
37 | 30, 33, 35, 36 | syl21anc 1317 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ {1}𝐵 = Σ𝑚 ∈ 𝐴 if(𝑚 ∈ {1}, 𝐵, 0)) |
38 | 11 | ralrimiva 2949 | . . . 4 ⊢ (𝜑 → ∀𝑚 ∈ 𝐴 𝐵 ∈ ℂ) |
39 | musumsum.1 | . . . . . 6 ⊢ (𝑚 = 1 → 𝐵 = 𝐶) | |
40 | 39 | eleq1d 2672 | . . . . 5 ⊢ (𝑚 = 1 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
41 | 40 | rspcv 3278 | . . . 4 ⊢ (1 ∈ 𝐴 → (∀𝑚 ∈ 𝐴 𝐵 ∈ ℂ → 𝐶 ∈ ℂ)) |
42 | 29, 38, 41 | sylc 63 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
43 | 39 | sumsn 14319 | . . 3 ⊢ ((1 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑚 ∈ {1}𝐵 = 𝐶) |
44 | 29, 42, 43 | syl2anc 691 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ {1}𝐵 = 𝐶) |
45 | 28, 37, 44 | 3eqtr2d 2650 | 1 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 ifcif 4036 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 ℕcn 10897 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 Σcsu 14264 ∥ cdvds 14821 μcmu 24621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-mu 24627 |
This theorem is referenced by: dchrmusum2 24983 dchrvmasum2lem 24985 mudivsum 25019 mulogsum 25021 mulog2sumlem2 25024 |
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