| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 3983 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ ℕ) |
| 3 | musum 27234 | . . . . . 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 7446 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = (if(𝑚 = 1, 1, 0) · 𝐵)) |
| 6 | fzfid 14014 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (1...𝑚) ∈ Fin) | |
| 7 | dvdsssfz1 16355 | . . . . . . 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 9301 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ∈ Fin) |
| 10 | musumsum.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
| 11 | elrabi 3687 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → 𝑘 ∈ ℕ) | |
| 12 | mucl 27184 | . . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (μ‘𝑘) ∈ ℤ) | |
| 13 | 11, 12 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℤ) |
| 14 | 13 | zcnd 12723 | . . . . . 6 ⊢ (𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} → (μ‘𝑘) ∈ ℂ) |
| 15 | 14 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝐴) ∧ 𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚}) → (μ‘𝑘) ∈ ℂ) |
| 16 | 9, 10, 15 | fsummulc1 15821 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} (μ‘𝑘) · 𝐵) = Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵)) |
| 17 | ovif 7531 | . . . . 5 ⊢ (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 = 1, (1 · 𝐵), (0 · 𝐵)) | |
| 18 | velsn 4642 | . . . . . . . . 9 ⊢ (𝑚 ∈ {1} ↔ 𝑚 = 1) | |
| 19 | 18 | bicomi 224 | . . . . . . . 8 ⊢ (𝑚 = 1 ↔ 𝑚 ∈ {1}) |
| 20 | 19 | a1i 11 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (𝑚 = 1 ↔ 𝑚 ∈ {1})) |
| 21 | mullid 11260 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (1 · 𝐵) = 𝐵) | |
| 22 | mul02 11439 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (0 · 𝐵) = 0) | |
| 23 | 20, 21, 22 | ifbieq12d 4554 | . . . . . 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 2789 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → (if(𝑚 = 1, 1, 0) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
| 26 | 5, 16, 25 | 3eqtr3d 2785 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝐴) → Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = if(𝑚 ∈ {1}, 𝐵, 0)) |
| 27 | 26 | sumeq2dv 15738 | . 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 4809 | . . 3 ⊢ (𝜑 → {1} ⊆ 𝐴) |
| 30 | 29 | sselda 3983 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝑚 ∈ 𝐴) |
| 31 | 30, 10 | syldan 591 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ {1}) → 𝐵 ∈ ℂ) |
| 32 | 31 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑚 ∈ {1}𝐵 ∈ ℂ) |
| 33 | musumsum.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
| 34 | 33 | olcd 875 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ (ℤ≥‘1) ∨ 𝐴 ∈ Fin)) |
| 35 | sumss2 15762 | . . 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 838 | . 2 ⊢ (𝜑 → Σ𝑚 ∈ {1}𝐵 = Σ𝑚 ∈ 𝐴 if(𝑚 ∈ {1}, 𝐵, 0)) |
| 37 | musumsum.1 | . . . . 5 ⊢ (𝑚 = 1 → 𝐵 = 𝐶) | |
| 38 | 37 | eleq1d 2826 | . . . 4 ⊢ (𝑚 = 1 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ)) |
| 39 | 10 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑚 ∈ 𝐴 𝐵 ∈ ℂ) |
| 40 | 38, 39, 28 | rspcdva 3623 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 41 | 37 | sumsn 15782 | . . 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 2783 | 1 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 Σ𝑘 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑚} ((μ‘𝑘) · 𝐵) = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 ifcif 4525 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 ℕcn 12266 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 Σcsu 15722 ∥ cdvds 16290 μcmu 27138 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-mu 27144 |
| This theorem is referenced by: dchrmusum2 27538 dchrvmasum2lem 27540 mudivsum 27574 mulogsum 27576 mulog2sumlem2 27579 |
| Copyright terms: Public domain | W3C validator |