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Mirrors > Home > MPE Home > Th. List > muval | Structured version Visualization version GIF version |
Description: The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.) |
Ref | Expression |
---|---|
muval | ⊢ (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4587 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑝↑2) ∥ 𝑥 ↔ (𝑝↑2) ∥ 𝐴)) | |
2 | 1 | rexbidv 3034 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴)) |
3 | breq2 4587 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐴)) | |
4 | 3 | rabbidv 3164 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} = {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) |
5 | 4 | fveq2d 6107 | . . . 4 ⊢ (𝑥 = 𝐴 → (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}) = (#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) |
6 | 5 | oveq2d 6565 | . . 3 ⊢ (𝑥 = 𝐴 → (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥})) = (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) |
7 | 2, 6 | ifbieq2d 4061 | . 2 ⊢ (𝑥 = 𝐴 → if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥}))) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
8 | df-mu 24627 | . 2 ⊢ μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥})))) | |
9 | c0ex 9913 | . . 3 ⊢ 0 ∈ V | |
10 | ovex 6577 | . . 3 ⊢ (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) ∈ V | |
11 | 9, 10 | ifex 4106 | . 2 ⊢ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) ∈ V |
12 | 7, 8, 11 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 ifcif 4036 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 -cneg 10146 ℕcn 10897 2c2 10947 ↑cexp 12722 #chash 12979 ∥ cdvds 14821 ℙcprime 15223 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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-mu 24627 |
This theorem is referenced by: muval1 24659 muval2 24660 isnsqf 24661 mule1 24674 |
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