fermltlchr - Metamath Proof Explorer

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Theorem fermltlchr 21388

Description: A generalization of Fermat's little theorem in a commutative ring 𝐹 of prime characteristic. See fermltl 16716. (Contributed by Thierry Arnoux, 9-Jan-2024.)

**Hypotheses**
Ref	Expression
fermltlchr.z	⊢ 𝑃 = (chr‘𝐹)
fermltlchr.b	⊢ 𝐵 = (Base‘𝐹)
fermltlchr.p	⊢ ↑ = (._g‘(mulGrp‘𝐹))
fermltlchr.1	⊢ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)
fermltlchr.2	⊢ (𝜑 → 𝑃 ∈ ℙ)
fermltlchr.3	⊢ (𝜑 → 𝐸 ∈ ℤ)
fermltlchr.4	⊢ (𝜑 → 𝐹 ∈ CRing)

**Assertion**
Ref	Expression
fermltlchr	⊢ (𝜑 → (𝑃 ↑ 𝐴) = 𝐴)

**Proof of Theorem fermltlchr**
Step	Hyp	Ref	Expression
1		fermltlchr.1	. 2 ⊢ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)
2		fermltlchr.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℙ)
3		prmnn 16608	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	nnnn0d 12529	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
5	2, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → 𝑃 ∈ ℕ₀)
7		fermltlchr.3	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℤ)
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → 𝐸 ∈ ℤ)
9		eqid 2724	. . . . . . . 8 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
10		zsscn 12563	. . . . . . . . 9 ⊢ ℤ ⊆ ℂ
11		eqid 2724	. . . . . . . . . 10 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
12		cnfldbas 21232	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂ_fld)
13	11, 12	mgpbas 20035	. . . . . . . . 9 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
14	10, 13	sseqtri 4010	. . . . . . . 8 ⊢ ℤ ⊆ (Base‘(mulGrp‘ℂ_fld))
15		eqid 2724	. . . . . . . 8 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
16		eqid 2724	. . . . . . . 8 ⊢ (inv_g‘(mulGrp‘ℂ_fld)) = (inv_g‘(mulGrp‘ℂ_fld))
17		cnring 21251	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
18	11	ringmgp 20134	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → (mulGrp‘ℂ_fld) ∈ Mnd)
19	17, 18	ax-mp 5	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) ∈ Mnd
20		cnfld1 21254	. . . . . . . . . . 11 ⊢ 1 = (1_r‘ℂ_fld)
21	11, 20	ringidval 20078	. . . . . . . . . 10 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
22		1z 12589	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
23	21, 22	eqeltrri 2822	. . . . . . . . 9 ⊢ (0_g‘(mulGrp‘ℂ_fld)) ∈ ℤ
24		eqid 2724	. . . . . . . . . 10 ⊢ (0_g‘(mulGrp‘ℂ_fld)) = (0_g‘(mulGrp‘ℂ_fld))
25	9, 13, 24	ress0g 18685	. . . . . . . . 9 ⊢ (((mulGrp‘ℂ_fld) ∈ Mnd ∧ (0_g‘(mulGrp‘ℂ_fld)) ∈ ℤ ∧ ℤ ⊆ ℂ) → (0_g‘(mulGrp‘ℂ_fld)) = (0_g‘((mulGrp‘ℂ_fld) ↾_s ℤ)))
26	19, 23, 10, 25	mp3an 1457	. . . . . . . 8 ⊢ (0_g‘(mulGrp‘ℂ_fld)) = (0_g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
27	9, 14, 15, 16, 26	ressmulgnn0 18995	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ₀ ∧ 𝐸 ∈ ℤ) → (𝑃(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐸) = (𝑃(._g‘(mulGrp‘ℂ_fld))𝐸))
28	6, 8, 27	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐸) = (𝑃(._g‘(mulGrp‘ℂ_fld))𝐸))
29	8	zcnd 12664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → 𝐸 ∈ ℂ)
30		cnfldexp 21262	. . . . . . 7 ⊢ ((𝐸 ∈ ℂ ∧ 𝑃 ∈ ℕ₀) → (𝑃(._g‘(mulGrp‘ℂ_fld))𝐸) = (𝐸↑𝑃))
31	29, 6, 30	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃(._g‘(mulGrp‘ℂ_fld))𝐸) = (𝐸↑𝑃))
32	28, 31	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐸) = (𝐸↑𝑃))
33	32	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → ((ℤRHom‘𝐹)‘(𝑃(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐸)) = ((ℤRHom‘𝐹)‘(𝐸↑𝑃)))
34		fermltlchr.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ CRing)
35	34	crngringd 20141	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ Ring)
36		eqid 2724	. . . . . . . . 9 ⊢ (ℤRHom‘𝐹) = (ℤRHom‘𝐹)
37	36	zrhrhm 21366	. . . . . . . 8 ⊢ (𝐹 ∈ Ring → (ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹))
38	35, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹))
39		zringmpg 21326	. . . . . . . 8 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
40		eqid 2724	. . . . . . . 8 ⊢ (mulGrp‘𝐹) = (mulGrp‘𝐹)
41	39, 40	rhmmhm 20371	. . . . . . 7 ⊢ ((ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹) → (ℤRHom‘𝐹) ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom (mulGrp‘𝐹)))
42	38, 41	syl 17	. . . . . 6 ⊢ (𝜑 → (ℤRHom‘𝐹) ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom (mulGrp‘𝐹)))
43	42	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (ℤRHom‘𝐹) ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom (mulGrp‘𝐹)))
44	9, 13	ressbas2 17181	. . . . . . 7 ⊢ (ℤ ⊆ ℂ → ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ)))
45	10, 44	ax-mp 5	. . . . . 6 ⊢ ℤ = (Base‘((mulGrp‘ℂ_fld) ↾_s ℤ))
46		eqid 2724	. . . . . 6 ⊢ (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ)) = (._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
47		fermltlchr.p	. . . . . 6 ⊢ ↑ = (._g‘(mulGrp‘𝐹))
48	45, 46, 47	mhmmulg 19032	. . . . 5 ⊢ (((ℤRHom‘𝐹) ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom (mulGrp‘𝐹)) ∧ 𝑃 ∈ ℕ₀ ∧ 𝐸 ∈ ℤ) → ((ℤRHom‘𝐹)‘(𝑃(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐸)) = (𝑃 ↑ ((ℤRHom‘𝐹)‘𝐸)))
49	43, 6, 8, 48	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → ((ℤRHom‘𝐹)‘(𝑃(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐸)) = (𝑃 ↑ ((ℤRHom‘𝐹)‘𝐸)))
50	7, 5	zexpcld 14050	. . . . . . . . 9 ⊢ (𝜑 → (𝐸↑𝑃) ∈ ℤ)
51		eqid 2724	. . . . . . . . . 10 ⊢ (-_g‘ℤ_ring) = (-_g‘ℤ_ring)
52	51	zringsubgval 21325	. . . . . . . . 9 ⊢ (((𝐸↑𝑃) ∈ ℤ ∧ 𝐸 ∈ ℤ) → ((𝐸↑𝑃) − 𝐸) = ((𝐸↑𝑃)(-_g‘ℤ_ring)𝐸))
53	50, 7, 52	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ((𝐸↑𝑃) − 𝐸) = ((𝐸↑𝑃)(-_g‘ℤ_ring)𝐸))
54	53	fveq2d 6885	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘((𝐸↑𝑃) − 𝐸)) = ((ℤRHom‘𝐹)‘((𝐸↑𝑃)(-_g‘ℤ_ring)𝐸)))
55	50	zred 12663	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸↑𝑃) ∈ ℝ)
56	7	zred 12663	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ)
57	2, 3	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ)
58	57	nnrpd 13011	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ ℝ⁺)
59		fermltl 16716	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐸 ∈ ℤ) → ((𝐸↑𝑃) mod 𝑃) = (𝐸 mod 𝑃))
60	2, 7, 59	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸↑𝑃) mod 𝑃) = (𝐸 mod 𝑃))
61		eqidd 2725	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 mod 𝑃) = (𝐸 mod 𝑃))
62	55, 56, 56, 56, 58, 60, 61	modsub12d 13890	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐸↑𝑃) − 𝐸) mod 𝑃) = ((𝐸 − 𝐸) mod 𝑃))
63		zcn 12560	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ ℤ → 𝐸 ∈ ℂ)
64	63	subidd 11556	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ ℤ → (𝐸 − 𝐸) = 0)
65	7, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 − 𝐸) = 0)
66	65	oveq1d 7416	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 − 𝐸) mod 𝑃) = (0 mod 𝑃))
67		0mod 13864	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℝ⁺ → (0 mod 𝑃) = 0)
68	58, 67	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0 mod 𝑃) = 0)
69	62, 66, 68	3eqtrd 2768	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸↑𝑃) − 𝐸) mod 𝑃) = 0)
70	50, 7	zsubcld 12668	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸↑𝑃) − 𝐸) ∈ ℤ)
71		dvdsval3 16198	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℕ ∧ ((𝐸↑𝑃) − 𝐸) ∈ ℤ) → (𝑃 ∥ ((𝐸↑𝑃) − 𝐸) ↔ (((𝐸↑𝑃) − 𝐸) mod 𝑃) = 0))
72	57, 70, 71	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∥ ((𝐸↑𝑃) − 𝐸) ↔ (((𝐸↑𝑃) − 𝐸) mod 𝑃) = 0))
73	69, 72	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∥ ((𝐸↑𝑃) − 𝐸))
74		fermltlchr.z	. . . . . . . . . 10 ⊢ 𝑃 = (chr‘𝐹)
75		eqid 2724	. . . . . . . . . 10 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
76	74, 36, 75	chrdvds 21385	. . . . . . . . 9 ⊢ ((𝐹 ∈ Ring ∧ ((𝐸↑𝑃) − 𝐸) ∈ ℤ) → (𝑃 ∥ ((𝐸↑𝑃) − 𝐸) ↔ ((ℤRHom‘𝐹)‘((𝐸↑𝑃) − 𝐸)) = (0_g‘𝐹)))
77	35, 70, 76	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∥ ((𝐸↑𝑃) − 𝐸) ↔ ((ℤRHom‘𝐹)‘((𝐸↑𝑃) − 𝐸)) = (0_g‘𝐹)))
78	73, 77	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘((𝐸↑𝑃) − 𝐸)) = (0_g‘𝐹))
79		rhmghm 20376	. . . . . . . . 9 ⊢ ((ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹) → (ℤRHom‘𝐹) ∈ (ℤ_ring GrpHom 𝐹))
80	38, 79	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℤRHom‘𝐹) ∈ (ℤ_ring GrpHom 𝐹))
81		zringbas 21308	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤ_ring)
82		eqid 2724	. . . . . . . . 9 ⊢ (-_g‘𝐹) = (-_g‘𝐹)
83	81, 51, 82	ghmsub 19139	. . . . . . . 8 ⊢ (((ℤRHom‘𝐹) ∈ (ℤ_ring GrpHom 𝐹) ∧ (𝐸↑𝑃) ∈ ℤ ∧ 𝐸 ∈ ℤ) → ((ℤRHom‘𝐹)‘((𝐸↑𝑃)(-_g‘ℤ_ring)𝐸)) = (((ℤRHom‘𝐹)‘(𝐸↑𝑃))(-_g‘𝐹)((ℤRHom‘𝐹)‘𝐸)))
84	80, 50, 7, 83	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘((𝐸↑𝑃)(-_g‘ℤ_ring)𝐸)) = (((ℤRHom‘𝐹)‘(𝐸↑𝑃))(-_g‘𝐹)((ℤRHom‘𝐹)‘𝐸)))
85	54, 78, 84	3eqtr3rd 2773	. . . . . 6 ⊢ (𝜑 → (((ℤRHom‘𝐹)‘(𝐸↑𝑃))(-_g‘𝐹)((ℤRHom‘𝐹)‘𝐸)) = (0_g‘𝐹))
86	34	crnggrpd 20142	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Grp)
87		eqid 2724	. . . . . . . . . 10 ⊢ (Base‘𝐹) = (Base‘𝐹)
88	81, 87	rhmf 20377	. . . . . . . . 9 ⊢ ((ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹) → (ℤRHom‘𝐹):ℤ⟶(Base‘𝐹))
89	38, 88	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℤRHom‘𝐹):ℤ⟶(Base‘𝐹))
90	89, 50	ffvelcdmd 7077	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘(𝐸↑𝑃)) ∈ (Base‘𝐹))
91	89, 7	ffvelcdmd 7077	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹))
92	87, 75, 82	grpsubeq0 18944	. . . . . . 7 ⊢ ((𝐹 ∈ Grp ∧ ((ℤRHom‘𝐹)‘(𝐸↑𝑃)) ∈ (Base‘𝐹) ∧ ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹)) → ((((ℤRHom‘𝐹)‘(𝐸↑𝑃))(-_g‘𝐹)((ℤRHom‘𝐹)‘𝐸)) = (0_g‘𝐹) ↔ ((ℤRHom‘𝐹)‘(𝐸↑𝑃)) = ((ℤRHom‘𝐹)‘𝐸)))
93	86, 90, 91, 92	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → ((((ℤRHom‘𝐹)‘(𝐸↑𝑃))(-_g‘𝐹)((ℤRHom‘𝐹)‘𝐸)) = (0_g‘𝐹) ↔ ((ℤRHom‘𝐹)‘(𝐸↑𝑃)) = ((ℤRHom‘𝐹)‘𝐸)))
94	85, 93	mpbid 231	. . . . 5 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘(𝐸↑𝑃)) = ((ℤRHom‘𝐹)‘𝐸))
95	94	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → ((ℤRHom‘𝐹)‘(𝐸↑𝑃)) = ((ℤRHom‘𝐹)‘𝐸))
96	33, 49, 95	3eqtr3d 2772	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃 ↑ ((ℤRHom‘𝐹)‘𝐸)) = ((ℤRHom‘𝐹)‘𝐸))
97		oveq2 7409	. . . 4 ⊢ (𝐴 = ((ℤRHom‘𝐹)‘𝐸) → (𝑃 ↑ 𝐴) = (𝑃 ↑ ((ℤRHom‘𝐹)‘𝐸)))
98	97	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃 ↑ 𝐴) = (𝑃 ↑ ((ℤRHom‘𝐹)‘𝐸)))
99		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → 𝐴 = ((ℤRHom‘𝐹)‘𝐸))
100	96, 98, 99	3eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃 ↑ 𝐴) = 𝐴)
101	1, 100	mpan2 688	1 ⊢ (𝜑 → (𝑃 ↑ 𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 class class class wbr 5138 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 0cc0 11106 1c1 11107 − cmin 11441 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 ℝ⁺crp 12971 mod cmo 13831 ↑cexp 14024 ∥ cdvds 16194 ℙcprime 16605 Basecbs 17143 ↾_s cress 17172 0_gc0g 17384 Mndcmnd 18657 MndHom cmhm 18701 Grpcgrp 18853 inv_gcminusg 18854 -_gcsg 18855 ._gcmg 18985 GrpHom cghm 19128 mulGrpcmgp 20029 Ringcrg 20128 CRingccrg 20129 RingHom crh 20361 ℂ_fldccnfld 21228 ℤ_ringczring 21301 ℤRHomczrh 21354 chrcchr 21356

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-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-dju 9892 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-xnn0 12542 df-z 12556 df-dec 12675 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-dvds 16195 df-gcd 16433 df-prm 16606 df-phi 16698 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-od 19438 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-subrng 20436 df-subrg 20461 df-drng 20579 df-cnfld 21229 df-zring 21302 df-zrh 21358 df-chr 21360

This theorem is referenced by: ply1fermltlchr 22153

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