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Mirrors > Home > MPE Home > Th. List > muval2 | Structured version Visualization version GIF version |
Description: The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.) |
Ref | Expression |
---|---|
muval2 | ⊢ ((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) → (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2939 | . . 3 ⊢ ((μ‘𝐴) ≠ 0 ↔ ¬ (μ‘𝐴) = 0) | |
2 | ifeqor 4582 | . . . . 5 ⊢ (if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 0 ∨ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) | |
3 | muval 27190 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) | |
4 | 3 | eqeq1d 2737 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 0)) |
5 | 3 | eqeq1d 2737 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) ↔ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
6 | 4, 5 | orbi12d 918 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (((μ‘𝐴) = 0 ∨ (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) ↔ (if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 0 ∨ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))))) |
7 | 2, 6 | mpbiri 258 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ∨ (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
8 | 7 | ord 864 | . . 3 ⊢ (𝐴 ∈ ℕ → (¬ (μ‘𝐴) = 0 → (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
9 | 1, 8 | biimtrid 242 | . 2 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 → (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
10 | 9 | imp 406 | 1 ⊢ ((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) → (μ‘𝐴) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 ifcif 4531 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 -cneg 11491 ℕcn 12264 2c2 12319 ↑cexp 14099 ♯chash 14366 ∥ cdvds 16287 ℙcprime 16705 μcmu 27153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-mu 27159 |
This theorem is referenced by: mumul 27239 musum 27249 |
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