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Mirrors > Home > MPE Home > Th. List > phival | Structured version Visualization version GIF version |
Description: Value of the Euler ϕ function. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
phival | ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7409 | . . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | oveq2 7409 | . . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥 gcd 𝑛) = (𝑥 gcd 𝑁)) | |
3 | 2 | eqeq1d 2726 | . . . 4 ⊢ (𝑛 = 𝑁 → ((𝑥 gcd 𝑛) = 1 ↔ (𝑥 gcd 𝑁) = 1)) |
4 | 1, 3 | rabeqbidv 3441 | . . 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 16695 | . 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 6988 | 1 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3424 ‘cfv 6533 (class class class)co 7401 1c1 11106 ℕcn 12208 ...cfz 13480 ♯chash 14286 gcd cgcd 16431 ϕcphi 16693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-phi 16695 |
This theorem is referenced by: phicl2 16697 phibnd 16700 dfphi2 16703 phiprmpw 16705 |
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