MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  muval Structured version   Visualization version   GIF version

Theorem muval 25715
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 5046 . . . 4 (𝑥 = 𝐴 → ((𝑝↑2) ∥ 𝑥 ↔ (𝑝↑2) ∥ 𝐴))
21rexbidv 3283 . . 3 (𝑥 = 𝐴 → (∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴))
3 breq2 5046 . . . . . 6 (𝑥 = 𝐴 → (𝑝𝑥𝑝𝐴))
43rabbidv 3455 . . . . 5 (𝑥 = 𝐴 → {𝑝 ∈ ℙ ∣ 𝑝𝑥} = {𝑝 ∈ ℙ ∣ 𝑝𝐴})
54fveq2d 6656 . . . 4 (𝑥 = 𝐴 → (♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}) = (♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))
65oveq2d 7156 . . 3 (𝑥 = 𝐴 → (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥})) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴})))
72, 6ifbieq2d 4464 . 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 25684 . 2 μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝑥}))))
9 c0ex 10624 . . 3 0 ∈ V
10 ovex 7173 . . 3 (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴})) ∈ V
119, 10ifex 4487 . 2 if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝𝐴}))) ∈ V
127, 8, 11fvmpt 6750 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 1538  wcel 2114  wrex 3131  {crab 3134  ifcif 4439   class class class wbr 5042  cfv 6334  (class class class)co 7140  0cc0 10526  1c1 10527  -cneg 10860  cn 11625  2c2 11680  cexp 13425  chash 13686  cdvds 15598  cprime 16004  μcmu 25678
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-mulcl 10588  ax-i2m1 10594
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-mu 25684
This theorem is referenced by:  muval1  25716  muval2  25717  isnsqf  25718  mule1  25731
  Copyright terms: Public domain W3C validator