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Theorem muval 26497
Description: The value of the MΓΆbius function. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
muval (𝐴 ∈ β„• β†’ (ΞΌβ€˜π΄) = if(βˆƒπ‘ ∈ β„™ (𝑝↑2) βˆ₯ 𝐴, 0, (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}))))
Distinct variable group:   𝐴,𝑝

Proof of Theorem muval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 breq2 5114 . . . 4 (π‘₯ = 𝐴 β†’ ((𝑝↑2) βˆ₯ π‘₯ ↔ (𝑝↑2) βˆ₯ 𝐴))
21rexbidv 3176 . . 3 (π‘₯ = 𝐴 β†’ (βˆƒπ‘ ∈ β„™ (𝑝↑2) βˆ₯ π‘₯ ↔ βˆƒπ‘ ∈ β„™ (𝑝↑2) βˆ₯ 𝐴))
3 breq2 5114 . . . . . 6 (π‘₯ = 𝐴 β†’ (𝑝 βˆ₯ π‘₯ ↔ 𝑝 βˆ₯ 𝐴))
43rabbidv 3418 . . . . 5 (π‘₯ = 𝐴 β†’ {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘₯} = {𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴})
54fveq2d 6851 . . . 4 (π‘₯ = 𝐴 β†’ (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘₯}) = (β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}))
65oveq2d 7378 . . 3 (π‘₯ = 𝐴 β†’ (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘₯})) = (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴})))
72, 6ifbieq2d 4517 . 2 (π‘₯ = 𝐴 β†’ if(βˆƒπ‘ ∈ β„™ (𝑝↑2) βˆ₯ π‘₯, 0, (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘₯}))) = if(βˆƒπ‘ ∈ β„™ (𝑝↑2) βˆ₯ 𝐴, 0, (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}))))
8 df-mu 26466 . 2 ΞΌ = (π‘₯ ∈ β„• ↦ if(βˆƒπ‘ ∈ β„™ (𝑝↑2) βˆ₯ π‘₯, 0, (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ π‘₯}))))
9 c0ex 11156 . . 3 0 ∈ V
10 ovex 7395 . . 3 (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴})) ∈ V
119, 10ifex 4541 . 2 if(βˆƒπ‘ ∈ β„™ (𝑝↑2) βˆ₯ 𝐴, 0, (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}))) ∈ V
127, 8, 11fvmpt 6953 1 (𝐴 ∈ β„• β†’ (ΞΌβ€˜π΄) = if(βˆƒπ‘ ∈ β„™ (𝑝↑2) βˆ₯ 𝐴, 0, (-1↑(β™―β€˜{𝑝 ∈ β„™ ∣ 𝑝 βˆ₯ 𝐴}))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074  {crab 3410  ifcif 4491   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  1c1 11059  -cneg 11393  β„•cn 12160  2c2 12215  β†‘cexp 13974  β™―chash 14237   βˆ₯ cdvds 16143  β„™cprime 16554  ΞΌcmu 26460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-mulcl 11120  ax-i2m1 11126
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-mu 26466
This theorem is referenced by:  muval1  26498  muval2  26499  isnsqf  26500  mule1  26513
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