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Theorem phival 15310
 Description: Value of the Euler ϕ function. (Contributed by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
phival (𝑁 ∈ ℕ → (ϕ‘𝑁) = (#‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
Distinct variable group:   𝑥,𝑁

Proof of Theorem phival
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . 4 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2 oveq2 6557 . . . . 5 (𝑛 = 𝑁 → (𝑥 gcd 𝑛) = (𝑥 gcd 𝑁))
32eqeq1d 2612 . . . 4 (𝑛 = 𝑁 → ((𝑥 gcd 𝑛) = 1 ↔ (𝑥 gcd 𝑁) = 1))
41, 3rabeqbidv 3168 . . 3 (𝑛 = 𝑁 → {𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1} = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
54fveq2d 6107 . 2 (𝑛 = 𝑁 → (#‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}) = (#‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
6 df-phi 15309 . 2 ϕ = (𝑛 ∈ ℕ ↦ (#‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
7 fvex 6113 . 2 (#‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) ∈ V
85, 6, 7fvmpt 6191 1 (𝑁 ∈ ℕ → (ϕ‘𝑁) = (#‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  ‘cfv 5804  (class class class)co 6549  1c1 9816  ℕcn 10897  ...cfz 12197  #chash 12979   gcd cgcd 15054  ϕcphi 15307 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 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-phi 15309 This theorem is referenced by:  phicl2  15311  phibnd  15314  dfphi2  15317  phiprmpw  15319
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