fermltlchr - Mathbox for Thierry Arnoux

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Theorem fermltlchr 32467

Description: A generalization of Fermat's little theorem in a commutative ring 𝐹 of prime characteristic. See fermltl 16714. (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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → 𝑃 ∈ ℕ₀)
7		fermltlchr.3	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℤ)
8	7	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → 𝐸 ∈ ℤ)
9		eqid 2733	. . . . . . . 8 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = ((mulGrp‘ℂ_fld) ↾_s ℤ)
10		zsscn 12563	. . . . . . . . 9 ⊢ ℤ ⊆ ℂ
11		eqid 2733	. . . . . . . . . 10 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
12		cnfldbas 20941	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂ_fld)
13	11, 12	mgpbas 19988	. . . . . . . . 9 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
14	10, 13	sseqtri 4018	. . . . . . . 8 ⊢ ℤ ⊆ (Base‘(mulGrp‘ℂ_fld))
15		eqid 2733	. . . . . . . 8 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
16		eqid 2733	. . . . . . . 8 ⊢ (inv_g‘(mulGrp‘ℂ_fld)) = (inv_g‘(mulGrp‘ℂ_fld))
17		cnring 20960	. . . . . . . . . 10 ⊢ ℂ_fld ∈ Ring
18	11	ringmgp 20056	. . . . . . . . . 10 ⊢ (ℂ_fld ∈ Ring → (mulGrp‘ℂ_fld) ∈ Mnd)
19	17, 18	ax-mp 5	. . . . . . . . 9 ⊢ (mulGrp‘ℂ_fld) ∈ Mnd
20		cnfld1 20963	. . . . . . . . . . 11 ⊢ 1 = (1_r‘ℂ_fld)
21	11, 20	ringidval 20001	. . . . . . . . . 10 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
22		1z 12589	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
23	21, 22	eqeltrri 2831	. . . . . . . . 9 ⊢ (0_g‘(mulGrp‘ℂ_fld)) ∈ ℤ
24		eqid 2733	. . . . . . . . . 10 ⊢ (0_g‘(mulGrp‘ℂ_fld)) = (0_g‘(mulGrp‘ℂ_fld))
25	9, 13, 24	ress0g 18650	. . . . . . . . 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 1462	. . . . . . . 8 ⊢ (0_g‘(mulGrp‘ℂ_fld)) = (0_g‘((mulGrp‘ℂ_fld) ↾_s ℤ))
27	9, 14, 15, 16, 26	ressmulgnn0 32173	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ₀ ∧ 𝐸 ∈ ℤ) → (𝑃(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐸) = (𝑃(._g‘(mulGrp‘ℂ_fld))𝐸))
28	6, 8, 27	syl2anc 585	. . . . . 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 20971	. . . . . . 7 ⊢ ((𝐸 ∈ ℂ ∧ 𝑃 ∈ ℕ₀) → (𝑃(._g‘(mulGrp‘ℂ_fld))𝐸) = (𝐸↑𝑃))
31	29, 6, 30	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃(._g‘(mulGrp‘ℂ_fld))𝐸) = (𝐸↑𝑃))
32	28, 31	eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃(._g‘((mulGrp‘ℂ_fld) ↾_s ℤ))𝐸) = (𝐸↑𝑃))
33	32	fveq2d 6893	. . . 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 20063	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ Ring)
36		eqid 2733	. . . . . . . . 9 ⊢ (ℤRHom‘𝐹) = (ℤRHom‘𝐹)
37	36	zrhrhm 21053	. . . . . . . 8 ⊢ (𝐹 ∈ Ring → (ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹))
38	35, 37	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹))
39		zringmpg 21033	. . . . . . . 8 ⊢ ((mulGrp‘ℂ_fld) ↾_s ℤ) = (mulGrp‘ℤ_ring)
40		eqid 2733	. . . . . . . 8 ⊢ (mulGrp‘𝐹) = (mulGrp‘𝐹)
41	39, 40	rhmmhm 20251	. . . . . . 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 482	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (ℤRHom‘𝐹) ∈ (((mulGrp‘ℂ_fld) ↾_s ℤ) MndHom (mulGrp‘𝐹)))
44	9, 13	ressbas2 17179	. . . . . . 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 2733	. . . . . 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 18990	. . . . 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 1372	. . . 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 2733	. . . . . . . . . 10 ⊢ (-_g‘ℤ_ring) = (-_g‘ℤ_ring)
52	51	zringsubgval 21032	. . . . . . . . 9 ⊢ (((𝐸↑𝑃) ∈ ℤ ∧ 𝐸 ∈ ℤ) → ((𝐸↑𝑃) − 𝐸) = ((𝐸↑𝑃)(-_g‘ℤ_ring)𝐸))
53	50, 7, 52	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → ((𝐸↑𝑃) − 𝐸) = ((𝐸↑𝑃)(-_g‘ℤ_ring)𝐸))
54	53	fveq2d 6893	. . . . . . 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 16714	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ 𝐸 ∈ ℤ) → ((𝐸↑𝑃) mod 𝑃) = (𝐸 mod 𝑃))
60	2, 7, 59	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸↑𝑃) mod 𝑃) = (𝐸 mod 𝑃))
61		eqidd 2734	. . . . . . . . . . 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 7421	. . . . . . . . . 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 2777	. . . . . . . . 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 585	. . . . . . . . 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 2733	. . . . . . . . . 10 ⊢ (0_g‘𝐹) = (0_g‘𝐹)
76	74, 36, 75	chrdvds 21072	. . . . . . . . 9 ⊢ ((𝐹 ∈ Ring ∧ ((𝐸↑𝑃) − 𝐸) ∈ ℤ) → (𝑃 ∥ ((𝐸↑𝑃) − 𝐸) ↔ ((ℤRHom‘𝐹)‘((𝐸↑𝑃) − 𝐸)) = (0_g‘𝐹)))
77	35, 70, 76	syl2anc 585	. . . . . . . 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 20255	. . . . . . . . 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 21016	. . . . . . . . 9 ⊢ ℤ = (Base‘ℤ_ring)
82		eqid 2733	. . . . . . . . 9 ⊢ (-_g‘𝐹) = (-_g‘𝐹)
83	81, 51, 82	ghmsub 19095	. . . . . . . 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 1372	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘((𝐸↑𝑃)(-_g‘ℤ_ring)𝐸)) = (((ℤRHom‘𝐹)‘(𝐸↑𝑃))(-_g‘𝐹)((ℤRHom‘𝐹)‘𝐸)))
85	54, 78, 84	3eqtr3rd 2782	. . . . . 6 ⊢ (𝜑 → (((ℤRHom‘𝐹)‘(𝐸↑𝑃))(-_g‘𝐹)((ℤRHom‘𝐹)‘𝐸)) = (0_g‘𝐹))
86	34	crnggrpd 20064	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Grp)
87		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘𝐹) = (Base‘𝐹)
88	81, 87	rhmf 20256	. . . . . . . . 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 7085	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘(𝐸↑𝑃)) ∈ (Base‘𝐹))
91	89, 7	ffvelcdmd 7085	. . . . . . 7 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹))
92	87, 75, 82	grpsubeq0 18906	. . . . . . 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 1372	. . . . . 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 482	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → ((ℤRHom‘𝐹)‘(𝐸↑𝑃)) = ((ℤRHom‘𝐹)‘𝐸))
96	33, 49, 95	3eqtr3d 2781	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃 ↑ ((ℤRHom‘𝐹)‘𝐸)) = ((ℤRHom‘𝐹)‘𝐸))
97		oveq2 7414	. . . 4 ⊢ (𝐴 = ((ℤRHom‘𝐹)‘𝐸) → (𝑃 ↑ 𝐴) = (𝑃 ↑ ((ℤRHom‘𝐹)‘𝐸)))
98	97	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃 ↑ 𝐴) = (𝑃 ↑ ((ℤRHom‘𝐹)‘𝐸)))
99		simpr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → 𝐴 = ((ℤRHom‘𝐹)‘𝐸))
100	96, 98, 99	3eqtr4d 2783	. 2 ⊢ ((𝜑 ∧ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)) → (𝑃 ↑ 𝐴) = 𝐴)
101	1, 100	mpan2 690	1 ⊢ (𝜑 → (𝑃 ↑ 𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 class class class wbr 5148 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 0cc0 11107 1c1 11108 − cmin 11441 ℕcn 12209 ℕ₀cn0 12469 ℤcz 12555 ℝ⁺crp 12971 mod cmo 13831 ↑cexp 14024 ∥ cdvds 16194 ℙcprime 16605 Basecbs 17141 ↾_s cress 17170 0_gc0g 17382 Mndcmnd 18622 MndHom cmhm 18666 Grpcgrp 18816 inv_gcminusg 18817 -_gcsg 18818 ._gcmg 18945 GrpHom cghm 19084 mulGrpcmgp 19982 Ringcrg 20050 CRingccrg 20051 RingHom crh 20241 ℂ_fldccnfld 20937 ℤ_ringczring 21010 ℤRHomczrh 21041 chrcchr 21043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-oadd 8467 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-dju 9893 df-card 9931 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 16696 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-0g 17384 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-grp 18819 df-minusg 18820 df-sbg 18821 df-mulg 18946 df-subg 18998 df-ghm 19085 df-od 19391 df-cmn 19645 df-mgp 19983 df-ur 20000 df-ring 20052 df-cring 20053 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-dvr 20208 df-rnghom 20244 df-drng 20310 df-subrg 20354 df-cnfld 20938 df-zring 21011 df-zrh 21045 df-chr 21047

This theorem is referenced by: ply1fermltlchr 32651

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