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Mirrors > Home > ILE Home > Th. List > phival | 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 | phivalfi 10967 | . . 3 ⊢ (𝑁 ∈ ℕ → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∈ Fin) | |
2 | hashcl 10023 | . . 3 ⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ∈ Fin → (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) ∈ ℕ0) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ ℕ → (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) ∈ ℕ0) |
4 | oveq2 5598 | . . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
5 | oveq2 5598 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑥 gcd 𝑛) = (𝑥 gcd 𝑁)) | |
6 | 5 | eqeq1d 2091 | . . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑥 gcd 𝑛) = 1 ↔ (𝑥 gcd 𝑁) = 1)) |
7 | 4, 6 | rabeqbidv 2607 | . . . 4 ⊢ (𝑛 = 𝑁 → {𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1} = {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) |
8 | 7 | fveq2d 5256 | . . 3 ⊢ (𝑛 = 𝑁 → (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1}) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
9 | df-phi 10966 | . . 3 ⊢ ϕ = (𝑛 ∈ ℕ ↦ (♯‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1})) | |
10 | 8, 9 | fvmptg 5324 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1}) ∈ ℕ0) → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
11 | 3, 10 | mpdan 412 | 1 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 {crab 2357 ‘cfv 4968 (class class class)co 5590 Fincfn 6386 1c1 7253 ℕcn 8315 ℕ0cn0 8564 ...cfz 9318 ♯chash 10017 gcd cgcd 10717 ϕcphi 10965 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-mulrcl 7346 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-mulass 7350 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-1rid 7354 ax-0id 7355 ax-rnegex 7356 ax-precex 7357 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 ax-pre-mulgt0 7364 ax-pre-mulext 7365 ax-arch 7366 ax-caucvg 7367 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fo 4974 df-f1o 4975 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-1st 5845 df-2nd 5846 df-recs 6001 df-frec 6087 df-1o 6112 df-er 6221 df-en 6387 df-dom 6388 df-fin 6389 df-sup 6585 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-reap 7951 df-ap 7958 df-div 8037 df-inn 8316 df-2 8374 df-3 8375 df-4 8376 df-n0 8565 df-z 8646 df-uz 8914 df-q 8999 df-rp 9029 df-fz 9319 df-fzo 9443 df-fl 9565 df-mod 9618 df-iseq 9740 df-iexp 9791 df-ihash 10018 df-cj 10102 df-re 10103 df-im 10104 df-rsqrt 10257 df-abs 10258 df-dvds 10576 df-gcd 10718 df-phi 10966 |
This theorem is referenced by: phicl2 10969 phibnd 10972 dfphi2 10975 phiprmpw 10977 |
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