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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 6557 | . . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | oveq2 6557 | . . . . 5 ⊢ (𝑛 = 𝑁 → (𝑥 gcd 𝑛) = (𝑥 gcd 𝑁)) | |
3 | 2 | eqeq1d 2612 | . . . 4 ⊢ (𝑛 = 𝑁 → ((𝑥 gcd 𝑛) = 1 ↔ (𝑥 gcd 𝑁) = 1)) |
4 | 1, 3 | rabeqbidv 3168 | . . 3 ⊢ (𝑛 = 𝑁 → {𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1} = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) |
5 | 4 | fveq2d 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 | |
8 | 5, 6, 7 | fvmpt 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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