Theorem phival 16832

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.

Step	Hyp	Ref	Expression
1		oveq2 7420	. . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2		oveq2 7420	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥 gcd 𝑛) = (𝑥 gcd 𝑁))
3	2	eqeq1d 2764	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑥 gcd 𝑛) = 1 ↔ (𝑥 gcd 𝑁) = 1))
4	1, 3	rabeqbidv 3433	. . 3 ⊢ (𝑛 = 𝑁 → {𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1} = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
5	4	fveq2d 6885	. 2 ⊢ (𝑛 = 𝑁 → (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
6		df-phi 16831	. 2 ⊢ ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
7		fvex 6894	. 2 ⊢ (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) ∈ V
8	5, 6, 7	fvmpt 6989	1 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  {crab 3415  ‘cfv 6536  (class class class)co 7412  1c1 11107  ℕcn 12239  ...cfz 13541  ♯chash 14373   gcd cgcd 16558  ϕcphi 16829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-phi 16831

This theorem is used by:  phicl2  16833  phibnd  16836  dfphi2  16839  phiprmpw  16841