fermltl - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > fermltl		GIF version

Theorem fermltl 12233

Description: Fermat's little theorem. When 𝑃 is prime, 𝐴↑𝑃≡𝐴 (mod 𝑃) for any 𝐴, see theorem 5.19 in [ApostolNT] p. 114. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 19-Mar-2022.)

**Assertion**
Ref	Expression
fermltl	⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))

**Proof of Theorem fermltl**
Step	Hyp	Ref	Expression
1		prmnn 12109	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
2		dvdsmodexp 11801	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ∈ ℕ ∧ 𝑃 ∥ 𝐴) → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))
3	2	3exp 1202	. . . 4 ⊢ (𝑃 ∈ ℕ → (𝑃 ∈ ℕ → (𝑃 ∥ 𝐴 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))))
4	1, 1, 3	sylc 62	. . 3 ⊢ (𝑃 ∈ ℙ → (𝑃 ∥ 𝐴 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃)))
5	4	adantr 276	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 ∥ 𝐴 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃)))
6		coprm 12143	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬ 𝑃 ∥ 𝐴 ↔ (𝑃 gcd 𝐴) = 1))
7		prmz 12110	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
8		gcdcom 11973	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑃 gcd 𝐴) = (𝐴 gcd 𝑃))
9	7, 8	sylan 283	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 gcd 𝐴) = (𝐴 gcd 𝑃))
10	9	eqeq1d 2186	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝑃 gcd 𝐴) = 1 ↔ (𝐴 gcd 𝑃) = 1))
11	6, 10	bitrd 188	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬ 𝑃 ∥ 𝐴 ↔ (𝐴 gcd 𝑃) = 1))
12		simp2 998	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 𝐴 ∈ ℤ)
13	1	3ad2ant1 1018	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 𝑃 ∈ ℕ)
14	13	phicld 12217	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (ϕ‘𝑃) ∈ ℕ)
15	14	nnnn0d 9228	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (ϕ‘𝑃) ∈ ℕ₀)
16		zexpcl 10534	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑃) ∈ ℕ₀) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ)
17	12, 15, 16	syl2anc 411	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ)
18		zq 9625	. . . . . . 7 ⊢ ((𝐴↑(ϕ‘𝑃)) ∈ ℤ → (𝐴↑(ϕ‘𝑃)) ∈ ℚ)
19	17, 18	syl 14	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (𝐴↑(ϕ‘𝑃)) ∈ ℚ)
20		1z 9278	. . . . . . 7 ⊢ 1 ∈ ℤ
21		zq 9625	. . . . . . 7 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
22	20, 21	mp1i 10	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 1 ∈ ℚ)
23		nnq 9632	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℚ)
24	13, 23	syl 14	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 𝑃 ∈ ℚ)
25	13	nngt0d 8962	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 0 < 𝑃)
26		eulerth 12232	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
27	1, 26	syl3an1 1271	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
28	19, 22, 12, 24, 25, 27	modqmul1 10376	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (((𝐴↑(ϕ‘𝑃)) · 𝐴) mod 𝑃) = ((1 · 𝐴) mod 𝑃))
29		phiprm 12222	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
30	29	3ad2ant1 1018	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (ϕ‘𝑃) = (𝑃 − 1))
31	30	oveq2d 5890	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (𝐴↑(ϕ‘𝑃)) = (𝐴↑(𝑃 − 1)))
32	31	oveq1d 5889	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) · 𝐴) = ((𝐴↑(𝑃 − 1)) · 𝐴))
33	12	zcnd 9375	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → 𝐴 ∈ ℂ)
34		expm1t 10547	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑃 ∈ ℕ) → (𝐴↑𝑃) = ((𝐴↑(𝑃 − 1)) · 𝐴))
35	33, 13, 34	syl2anc 411	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (𝐴↑𝑃) = ((𝐴↑(𝑃 − 1)) · 𝐴))
36	32, 35	eqtr4d 2213	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) · 𝐴) = (𝐴↑𝑃))
37	36	oveq1d 5889	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (((𝐴↑(ϕ‘𝑃)) · 𝐴) mod 𝑃) = ((𝐴↑𝑃) mod 𝑃))
38	33	mulid2d 7975	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → (1 · 𝐴) = 𝐴)
39	38	oveq1d 5889	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((1 · 𝐴) mod 𝑃) = (𝐴 mod 𝑃))
40	28, 37, 39	3eqtr3d 2218	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))
41	40	3expia 1205	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝑃) = 1 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃)))
42	11, 41	sylbid 150	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬ 𝑃 ∥ 𝐴 → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃)))
43		dvdsdc 11804	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℤ) → DECID 𝑃 ∥ 𝐴)
44	1, 43	sylan 283	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → DECID 𝑃 ∥ 𝐴)
45		exmiddc 836	. . 3 ⊢ (DECID 𝑃 ∥ 𝐴 → (𝑃 ∥ 𝐴 ∨ ¬ 𝑃 ∥ 𝐴))
46	44, 45	syl 14	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 ∥ 𝐴 ∨ ¬ 𝑃 ∥ 𝐴))
47	5, 42, 46	mpjaod 718	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → ((𝐴↑𝑃) mod 𝑃) = (𝐴 mod 𝑃))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 DECID wdc 834 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4003 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 1c1 7811 · cmul 7815 − cmin 8127 ℕcn 8918 ℕ₀cn0 9175 ℤcz 9252 ℚcq 9618 mod cmo 10321 ↑cexp 10518 ∥ cdvds 11793 gcd cgcd 11942 ℙcprime 12106 ϕcphi 12208

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-2o 6417 df-oadd 6420 df-er 6534 df-en 6740 df-dom 6741 df-fin 6742 df-sup 6982 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-rp 9653 df-fz 10008 df-fzo 10142 df-fl 10269 df-mod 10322 df-seqfrec 10445 df-exp 10519 df-ihash 10755 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 df-clim 11286 df-proddc 11558 df-dvds 11794 df-gcd 11943 df-prm 12107 df-phi 12210

This theorem is referenced by: (None)

W3C validator