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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 3944 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ ℕ) |
3 | musum 26540 | . . . . . 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 7372 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = (if(𝑚 = 1, 1, 0) · 𝐵)) |
6 | fzfid 13878 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (1...𝑚) ∈ Fin) | |
7 | dvdsssfz1 16200 | . . . . . . 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 9211 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ∈ Fin) |
10 | musumsum.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
11 | elrabi 3639 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → 𝑘 ∈ ℕ) | |
12 | mucl 26490 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ) | |
13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℤ) |
14 | 13 | zcnd 12608 | . . . . . 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 15670 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵)) |
17 | ovif 7454 | . . . . 5 ⊢ (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) | |
18 | velsn 4602 | . . . . . . . . 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 11154 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
22 | mul02 11333 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (0 · 𝐵) = 0) | |
23 | 20, 21, 22 | ifbieq12d 4514 | . . . . . 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 2788 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
26 | 5, 16, 25 | 3eqtr3d 2784 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
27 | 26 | sumeq2dv 15588 | . 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 4769 | . . 3 ⊢ (𝜑 → {1} ⊆ 𝐴) |
30 | 29 | sselda 3944 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝑚 ∈ 𝐴) |
31 | 30, 10 | syldan 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝐵 ∈ ℂ) |
32 | 31 | ralrimiva 3143 | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ {1}𝐵 ∈ ℂ) |
33 | musumsum.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
34 | 33 | olcd 872 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ (ℤ≥‘1) ∨ 𝐴 ∈ Fin)) |
35 | sumss2 15611 | . . 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 2822 | . . . 4 ⊢ (𝑚 = 1 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
39 | 10 | ralrimiva 3143 | . . . 4 ⊢ (𝜑 → ∀𝑚 ∈ 𝐴 𝐵 ∈ ℂ) |
40 | 38, 39, 28 | rspcdva 3582 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
41 | 37 | sumsn 15631 | . . 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 2782 | 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 845 = wceq 1541 ∈ wcel 2106 ∀wral 3064 {crab 3407 ⊆ wss 3910 ifcif 4486 {csn 4586 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 0cc0 11051 1c1 11052 · cmul 11056 ℕcn 12153 ℤcz 12499 ℤ≥cuz 12763 ...cfz 13424 Σcsu 15570 ∥ cdvds 16136 μcmu 26444 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-dvds 16137 df-gcd 16375 df-prm 16548 df-pc 16709 df-mu 26450 |
This theorem is referenced by: dchrmusum2 26842 dchrvmasum2lem 26844 mudivsum 26878 mulogsum 26880 mulog2sumlem2 26883 |
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