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Theorem phival 15960
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 6984 . . . 4 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2 oveq2 6984 . . . . 5 (𝑛 = 𝑁 → (𝑥 gcd 𝑛) = (𝑥 gcd 𝑁))
32eqeq1d 2781 . . . 4 (𝑛 = 𝑁 → ((𝑥 gcd 𝑛) = 1 ↔ (𝑥 gcd 𝑁) = 1))
41, 3rabeqbidv 3409 . . 3 (𝑛 = 𝑁 → {𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1} = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
54fveq2d 6503 . 2 (𝑛 = 𝑁 → (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
6 df-phi 15959 . 2 ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
7 fvex 6512 . 2 (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) ∈ V
85, 6, 7fvmpt 6595 1 (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  {crab 3093  cfv 6188  (class class class)co 6976  1c1 10336  cn 11439  ...cfz 12708  chash 13505   gcd cgcd 15703  ϕcphi 15957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-phi 15959
This theorem is referenced by:  phicl2  15961  phibnd  15964  dfphi2  15967  phiprmpw  15969
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