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Theorem znfld 21108

Description: The ℤ/nℤ structure is a finite field when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)

**Hypothesis**
Ref	Expression
zntos.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)

**Assertion**
Ref	Expression
znfld	⊢ (𝑁 ∈ ℙ → 𝑌 ∈ Field)

Proof of Theorem znfld Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmnn 16608	. . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
2		nnnn0 12476	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3	1, 2	syl 17	. . . 4 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ₀)
4		zntos.y	. . . . 5 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
5	4	zncrng 21092	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 ∈ CRing)
6	3, 5	syl 17	. . 3 ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ CRing)
7		crngring 20062	. . . . . 6 ⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring)
8	1, 2, 5, 7	4syl 19	. . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ Ring)
9		hash2 14362	. . . . . . 7 ⊢ (♯‘2_o) = 2
10		prmuz2 16630	. . . . . . . . 9 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ_≥‘2))
11		eluzle 12832	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ≤ 𝑁)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ ℙ → 2 ≤ 𝑁)
13		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘𝑌) = (Base‘𝑌)
14	4, 13	znhash 21106	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (♯‘(Base‘𝑌)) = 𝑁)
15	1, 14	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ ℙ → (♯‘(Base‘𝑌)) = 𝑁)
16	12, 15	breqtrrd 5176	. . . . . . 7 ⊢ (𝑁 ∈ ℙ → 2 ≤ (♯‘(Base‘𝑌)))
17	9, 16	eqbrtrid 5183	. . . . . 6 ⊢ (𝑁 ∈ ℙ → (♯‘2_o) ≤ (♯‘(Base‘𝑌)))
18		2onn 8638	. . . . . . . 8 ⊢ 2_o ∈ ω
19		nnfi 9164	. . . . . . . 8 ⊢ (2_o ∈ ω → 2_o ∈ Fin)
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ 2_o ∈ Fin
21		fvex 6902	. . . . . . 7 ⊢ (Base‘𝑌) ∈ V
22		hashdom 14336	. . . . . . 7 ⊢ ((2_o ∈ Fin ∧ (Base‘𝑌) ∈ V) → ((♯‘2_o) ≤ (♯‘(Base‘𝑌)) ↔ 2_o ≼ (Base‘𝑌)))
23	20, 21, 22	mp2an 691	. . . . . 6 ⊢ ((♯‘2_o) ≤ (♯‘(Base‘𝑌)) ↔ 2_o ≼ (Base‘𝑌))
24	17, 23	sylib 217	. . . . 5 ⊢ (𝑁 ∈ ℙ → 2_o ≼ (Base‘𝑌))
25	13	isnzr2 20290	. . . . 5 ⊢ (𝑌 ∈ NzRing ↔ (𝑌 ∈ Ring ∧ 2_o ≼ (Base‘𝑌)))
26	8, 24, 25	sylanbrc 584	. . . 4 ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ NzRing)
27		eqid 2733	. . . . . . . . 9 ⊢ (ℤRHom‘𝑌) = (ℤRHom‘𝑌)
28	4, 13, 27	znzrhfo 21095	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌))
29	3, 28	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ ℙ → (ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌))
30		foelrn 7105	. . . . . . . 8 ⊢ (((ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌) ∧ 𝑥 ∈ (Base‘𝑌)) → ∃𝑧 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑧))
31		foelrn 7105	. . . . . . . 8 ⊢ (((ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌)) → ∃𝑤 ∈ ℤ 𝑦 = ((ℤRHom‘𝑌)‘𝑤))
32	30, 31	anim12dan 620	. . . . . . 7 ⊢ (((ℤRHom‘𝑌):ℤ–onto→(Base‘𝑌) ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌))) → (∃𝑧 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ ∃𝑤 ∈ ℤ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)))
33	29, 32	sylan 581	. . . . . 6 ⊢ ((𝑁 ∈ ℙ ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌))) → (∃𝑧 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ ∃𝑤 ∈ ℤ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)))
34		reeanv 3227	. . . . . . . 8 ⊢ (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)) ↔ (∃𝑧 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ ∃𝑤 ∈ ℤ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)))
35		euclemma 16647	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℙ ∧ 𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (𝑁 ∥ (𝑧 · 𝑤) ↔ (𝑁 ∥ 𝑧 ∨ 𝑁 ∥ 𝑤)))
36	35	3expb 1121	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (𝑁 ∥ (𝑧 · 𝑤) ↔ (𝑁 ∥ 𝑧 ∨ 𝑁 ∥ 𝑤)))
37	8	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → 𝑌 ∈ Ring)
38	27	zrhrhm 21053	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ Ring → (ℤRHom‘𝑌) ∈ (ℤ_ring RingHom 𝑌))
39	37, 38	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (ℤRHom‘𝑌) ∈ (ℤ_ring RingHom 𝑌))
40		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → 𝑧 ∈ ℤ)
41		simprr 772	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → 𝑤 ∈ ℤ)
42		zringbas 21016	. . . . . . . . . . . . . . . 16 ⊢ ℤ = (Base‘ℤ_ring)
43		zringmulr 21019	. . . . . . . . . . . . . . . 16 ⊢ · = (._r‘ℤ_ring)
44		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (._r‘𝑌) = (._r‘𝑌)
45	42, 43, 44	rhmmul 20257	. . . . . . . . . . . . . . 15 ⊢ (((ℤRHom‘𝑌) ∈ (ℤ_ring RingHom 𝑌) ∧ 𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ) → ((ℤRHom‘𝑌)‘(𝑧 · 𝑤)) = (((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)))
46	39, 40, 41, 45	syl3anc 1372	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → ((ℤRHom‘𝑌)‘(𝑧 · 𝑤)) = (((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)))
47	46	eqeq1d 2735	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (((ℤRHom‘𝑌)‘(𝑧 · 𝑤)) = (0_g‘𝑌) ↔ (((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)) = (0_g‘𝑌)))
48		zmulcl 12608	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ) → (𝑧 · 𝑤) ∈ ℤ)
49		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑌) = (0_g‘𝑌)
50	4, 27, 49	zndvds0 21098	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑧 · 𝑤) ∈ ℤ) → (((ℤRHom‘𝑌)‘(𝑧 · 𝑤)) = (0_g‘𝑌) ↔ 𝑁 ∥ (𝑧 · 𝑤)))
51	3, 48, 50	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (((ℤRHom‘𝑌)‘(𝑧 · 𝑤)) = (0_g‘𝑌) ↔ 𝑁 ∥ (𝑧 · 𝑤)))
52	47, 51	bitr3d 281	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → ((((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)) = (0_g‘𝑌) ↔ 𝑁 ∥ (𝑧 · 𝑤)))
53	4, 27, 49	zndvds0 21098	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑧 ∈ ℤ) → (((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ↔ 𝑁 ∥ 𝑧))
54	3, 40, 53	syl2an2r 684	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ↔ 𝑁 ∥ 𝑧))
55	4, 27, 49	zndvds0 21098	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑤 ∈ ℤ) → (((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌) ↔ 𝑁 ∥ 𝑤))
56	3, 41, 55	syl2an2r 684	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → (((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌) ↔ 𝑁 ∥ 𝑤))
57	54, 56	orbi12d 918	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → ((((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ∨ ((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌)) ↔ (𝑁 ∥ 𝑧 ∨ 𝑁 ∥ 𝑤)))
58	36, 52, 57	3bitr4d 311	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → ((((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)) = (0_g‘𝑌) ↔ (((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ∨ ((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌))))
59	58	biimpd 228	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → ((((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)) = (0_g‘𝑌) → (((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ∨ ((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌))))
60		oveq12 7415	. . . . . . . . . . . 12 ⊢ ((𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)) → (𝑥(._r‘𝑌)𝑦) = (((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)))
61	60	eqeq1d 2735	. . . . . . . . . . 11 ⊢ ((𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)) → ((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) ↔ (((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)) = (0_g‘𝑌)))
62		eqeq1 2737	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((ℤRHom‘𝑌)‘𝑧) → (𝑥 = (0_g‘𝑌) ↔ ((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌)))
63	62	orbi1d 916	. . . . . . . . . . . 12 ⊢ (𝑥 = ((ℤRHom‘𝑌)‘𝑧) → ((𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌)) ↔ (((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌))))
64		eqeq1 2737	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((ℤRHom‘𝑌)‘𝑤) → (𝑦 = (0_g‘𝑌) ↔ ((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌)))
65	64	orbi2d 915	. . . . . . . . . . . 12 ⊢ (𝑦 = ((ℤRHom‘𝑌)‘𝑤) → ((((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌)) ↔ (((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ∨ ((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌))))
66	63, 65	sylan9bb 511	. . . . . . . . . . 11 ⊢ ((𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)) → ((𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌)) ↔ (((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ∨ ((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌))))
67	61, 66	imbi12d 345	. . . . . . . . . 10 ⊢ ((𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)) → (((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) → (𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌))) ↔ ((((ℤRHom‘𝑌)‘𝑧)(._r‘𝑌)((ℤRHom‘𝑌)‘𝑤)) = (0_g‘𝑌) → (((ℤRHom‘𝑌)‘𝑧) = (0_g‘𝑌) ∨ ((ℤRHom‘𝑌)‘𝑤) = (0_g‘𝑌)))))
68	59, 67	syl5ibrcom 246	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑤 ∈ ℤ)) → ((𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)) → ((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) → (𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌)))))
69	68	rexlimdvva 3212	. . . . . . . 8 ⊢ (𝑁 ∈ ℙ → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)) → ((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) → (𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌)))))
70	34, 69	biimtrrid 242	. . . . . . 7 ⊢ (𝑁 ∈ ℙ → ((∃𝑧 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ ∃𝑤 ∈ ℤ 𝑦 = ((ℤRHom‘𝑌)‘𝑤)) → ((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) → (𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌)))))
71	70	imp 408	. . . . . 6 ⊢ ((𝑁 ∈ ℙ ∧ (∃𝑧 ∈ ℤ 𝑥 = ((ℤRHom‘𝑌)‘𝑧) ∧ ∃𝑤 ∈ ℤ 𝑦 = ((ℤRHom‘𝑌)‘𝑤))) → ((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) → (𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌))))
72	33, 71	syldan 592	. . . . 5 ⊢ ((𝑁 ∈ ℙ ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌))) → ((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) → (𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌))))
73	72	ralrimivva 3201	. . . 4 ⊢ (𝑁 ∈ ℙ → ∀𝑥 ∈ (Base‘𝑌)∀𝑦 ∈ (Base‘𝑌)((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) → (𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌))))
74	13, 44, 49	isdomn 20903	. . . 4 ⊢ (𝑌 ∈ Domn ↔ (𝑌 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑌)∀𝑦 ∈ (Base‘𝑌)((𝑥(._r‘𝑌)𝑦) = (0_g‘𝑌) → (𝑥 = (0_g‘𝑌) ∨ 𝑦 = (0_g‘𝑌)))))
75	26, 73, 74	sylanbrc 584	. . 3 ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ Domn)
76		isidom 20915	. . 3 ⊢ (𝑌 ∈ IDomn ↔ (𝑌 ∈ CRing ∧ 𝑌 ∈ Domn))
77	6, 75, 76	sylanbrc 584	. 2 ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ IDomn)
78	4, 13	znfi 21107	. . . 4 ⊢ (𝑁 ∈ ℕ → (Base‘𝑌) ∈ Fin)
79	1, 78	syl 17	. . 3 ⊢ (𝑁 ∈ ℙ → (Base‘𝑌) ∈ Fin)
80	13	fiidomfld 20920	. . 3 ⊢ ((Base‘𝑌) ∈ Fin → (𝑌 ∈ IDomn ↔ 𝑌 ∈ Field))
81	79, 80	syl 17	. 2 ⊢ (𝑁 ∈ ℙ → (𝑌 ∈ IDomn ↔ 𝑌 ∈ Field))
82	77, 81	mpbid 231	1 ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ Field)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 class class class wbr 5148 –onto→wfo 6539 ‘cfv 6541 (class class class)co 7406 ωcom 7852 2_oc2o 8457 ≼ cdom 8934 Fincfn 8936 · cmul 11112 ≤ cle 11246 ℕcn 12209 2c2 12264 ℕ₀cn0 12469 ℤcz 12555 ℤ_≥cuz 12819 ♯chash 14287 ∥ cdvds 16194 ℙcprime 16605 Basecbs 17141 ._rcmulr 17195 0_gc0g 17382 Ringcrg 20050 CRingccrg 20051 RingHom crh 20241 NzRingcnzr 20284 Fieldcfield 20309 Domncdomn 20889 IDomncidom 20890 ℤ_ringczring 21010 ℤRHomczrh 21041 ℤ/nℤczn 21044

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-ec 8702 df-qs 8706 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-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-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-0g 17384 df-imas 17451 df-qus 17452 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-nsg 18999 df-eqg 19000 df-ghm 19085 df-cmn 19645 df-abl 19646 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-rnghom 20244 df-nzr 20285 df-drng 20310 df-field 20311 df-subrg 20354 df-lmod 20466 df-lss 20536 df-lsp 20576 df-sra 20778 df-rgmod 20779 df-lidl 20780 df-rsp 20781 df-2idl 20850 df-rlreg 20892 df-domn 20893 df-idom 20894 df-cnfld 20938 df-zring 21011 df-zrh 21045 df-zn 21048

This theorem is referenced by: znidomb 21109 lgsqrlem1 26839 lgsqrlem2 26840 lgsqrlem3 26841 lgsqrlem4 26842 lgseisenlem3 26870 lgseisenlem4 26871

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