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Theorem muval 25636
Description: The value of the Möbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
muval (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))))
Distinct variable group:   𝐴,𝑝

Proof of Theorem muval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq2 5061 . . . 4 (𝑥 = 𝐴 → ((𝑝↑2) ∥ 𝑥 ↔ (𝑝↑2) ∥ 𝐴))
21rexbidv 3294 . . 3 (𝑥 = 𝐴 → (∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴))
3 breq2 5061 . . . . . 6 (𝑥 = 𝐴 → (𝑝𝑥𝑝𝐴))
43rabbidv 3478 . . . . 5 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = {𝑝 ∈ ℙ ∣ 𝑝𝐴})
54fveq2d 6667 . . . 4 (𝑥 = 𝐴 → (♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}) = (♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))
65oveq2d 7161 . . 3 (𝑥 = 𝐴 → (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥})) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴})))
72, 6ifbieq2d 4488 . 2 (𝑥 = 𝐴 → if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))))
8 df-mu 25605 . 2 μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))
9 c0ex 10623 . . 3 0 ∈ V
10 ovex 7178 . . 3 (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴})) ∈ V
119, 10ifex 4511 . 2 if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))) ∈ V
127, 8, 11fvmpt 6761 1 (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wrex 3136  {crab 3139  ifcif 4463   class class class wbr 5057  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526  -cneg 10859  cn 11626  2c2 11680  cexp 13417  chash 13678  cdvds 15595  cprime 16003  μcmu 25599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-mu 25605
This theorem is referenced by:  muval1  25637  muval2  25638  isnsqf  25639  mule1  25652
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