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Mirrors > Home > ILE Home > Th. List > eulerth | GIF version |
Description: Euler's theorem, a generalization of Fermat's little theorem. If 𝐴 and 𝑁 are coprime, then 𝐴↑ϕ(𝑁)≡1 (mod 𝑁). This is Metamath 100 proof #10. Also called Euler-Fermat theorem, see theorem 5.17 in [ApostolNT] p. 113. (Contributed by Mario Carneiro, 28-Feb-2014.) |
Ref | Expression |
---|---|
eulerth | ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phicl 12353 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | |
2 | 1 | nnnn0d 9293 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ0) |
3 | hashfz1 10854 | . . . . . . 7 ⊢ ((ϕ‘𝑁) ∈ ℕ0 → (♯‘(1...(ϕ‘𝑁))) = (ϕ‘𝑁)) | |
4 | 2, 3 | syl 14 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (♯‘(1...(ϕ‘𝑁))) = (ϕ‘𝑁)) |
5 | dfphi2 12358 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) = (♯‘{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1})) | |
6 | 4, 5 | eqtrd 2226 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (♯‘(1...(ϕ‘𝑁))) = (♯‘{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1})) |
7 | 6 | 3ad2ant1 1020 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (♯‘(1...(ϕ‘𝑁))) = (♯‘{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1})) |
8 | 1zzd 9344 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 1 ∈ ℤ) | |
9 | 1 | 3ad2ant1 1020 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ϕ‘𝑁) ∈ ℕ) |
10 | 9 | nnzd 9438 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ϕ‘𝑁) ∈ ℤ) |
11 | 8, 10 | fzfigd 10502 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (1...(ϕ‘𝑁)) ∈ Fin) |
12 | id 19 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) | |
13 | oveq1 5925 | . . . . . . . 8 ⊢ (𝑘 = 𝑦 → (𝑘 gcd 𝑁) = (𝑦 gcd 𝑁)) | |
14 | 13 | eqeq1d 2202 | . . . . . . 7 ⊢ (𝑘 = 𝑦 → ((𝑘 gcd 𝑁) = 1 ↔ (𝑦 gcd 𝑁) = 1)) |
15 | 14 | cbvrabv 2759 | . . . . . 6 ⊢ {𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1} = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} |
16 | 12, 15 | eulerthlemfi 12366 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → {𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1} ∈ Fin) |
17 | hashen 10855 | . . . . 5 ⊢ (((1...(ϕ‘𝑁)) ∈ Fin ∧ {𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1} ∈ Fin) → ((♯‘(1...(ϕ‘𝑁))) = (♯‘{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) ↔ (1...(ϕ‘𝑁)) ≈ {𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1})) | |
18 | 11, 16, 17 | syl2anc 411 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((♯‘(1...(ϕ‘𝑁))) = (♯‘{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) ↔ (1...(ϕ‘𝑁)) ≈ {𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1})) |
19 | 7, 18 | mpbid 147 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (1...(ϕ‘𝑁)) ≈ {𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) |
20 | bren 6801 | . . 3 ⊢ ((1...(ϕ‘𝑁)) ≈ {𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1} ↔ ∃𝑓 𝑓:(1...(ϕ‘𝑁))–1-1-onto→{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) | |
21 | 19, 20 | sylib 122 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ∃𝑓 𝑓:(1...(ϕ‘𝑁))–1-1-onto→{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) |
22 | simpl 109 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑓:(1...(ϕ‘𝑁))–1-1-onto→{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) | |
23 | simpr 110 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑓:(1...(ϕ‘𝑁))–1-1-onto→{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) → 𝑓:(1...(ϕ‘𝑁))–1-1-onto→{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) | |
24 | 22, 15, 23 | eulerthlemth 12370 | . 2 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑓:(1...(ϕ‘𝑁))–1-1-onto→{𝑘 ∈ (0..^𝑁) ∣ (𝑘 gcd 𝑁) = 1}) → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁)) |
25 | 21, 24 | exlimddv 1910 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∃wex 1503 ∈ wcel 2164 {crab 2476 class class class wbr 4029 –1-1-onto→wf1o 5253 ‘cfv 5254 (class class class)co 5918 ≈ cen 6792 Fincfn 6794 0cc0 7872 1c1 7873 ℕcn 8982 ℕ0cn0 9240 ℤcz 9317 ...cfz 10074 ..^cfzo 10208 mod cmo 10393 ↑cexp 10609 ♯chash 10846 gcd cgcd 12079 ϕcphi 12347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-fl 10339 df-mod 10394 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-proddc 11694 df-dvds 11931 df-gcd 12080 df-phi 12349 |
This theorem is referenced by: fermltl 12372 prmdiv 12373 odzcllem 12380 odzphi 12384 vfermltl 12389 lgslem1 15116 |
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