Theorem phival 16732

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 7368	. . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2		oveq2 7368	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥 gcd 𝑛) = (𝑥 gcd 𝑁))
3	2	eqeq1d 2743	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑥 gcd 𝑛) = 1 ↔ (𝑥 gcd 𝑁) = 1))
4	1, 3	rabeqbidv 3411	. . 3 ⊢ (𝑛 = 𝑁 → {𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1} = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})
5	4	fveq2d 6835	. 2 ⊢ (𝑛 = 𝑁 → (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))
6		df-phi 16731	. 2 ⊢ ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}))
7		fvex 6844	. 2 ⊢ (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) ∈ V
8	5, 6, 7	fvmpt 6939	1 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  {crab 3393  ‘cfv 6489  (class class class)co 7360  1c1 11034  ℕcn 12169  ...cfz 13456  ♯chash 14287   gcd cgcd 16458  ϕcphi 16729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7363  df-phi 16731

This theorem is referenced by:  phicl2  16733  phibnd  16736  dfphi2  16739  phiprmpw  16741