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Theorem dchrfi 27206

Description: The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrabl.g	⊢ 𝐺 = (DChr‘𝑁)
dchrfi.b	⊢ 𝐷 = (Base‘𝐺)

**Assertion**
Ref	Expression
dchrfi	⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)

Proof of Theorem dchrfi Dummy variables 𝑥 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snfi 8979	. . . 4 ⊢ {0} ∈ Fin
2		cnex 11108	. . . . . . . . 9 ⊢ ℂ ∈ V
3	2	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ℂ ∈ V)
4		ovexd 7391	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → (𝑧↑(ϕ‘𝑁)) ∈ V)
5		1cnd 11128	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
6		eqidd 2736	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) = (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))))
7		fconstmpt 5682	. . . . . . . . 9 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
8	7	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1))
9	3, 4, 5, 6, 8	offval2 7640	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_f − (ℂ × {1})) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)))
10		ssid 3939	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ℂ ⊆ ℂ)
12		1cnd 11128	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
13		phicl 16728	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
14	13	nnnn0d 12487	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ₀)
15		plypow 26158	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
16	11, 12, 14, 15	syl3anc 1374	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
17		ax-1cn 11085	. . . . . . . . 9 ⊢ 1 ∈ ℂ
18		plyconst 26159	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → (ℂ × {1}) ∈ (Poly‘ℂ))
19	10, 17, 18	mp2an 693	. . . . . . . 8 ⊢ (ℂ × {1}) ∈ (Poly‘ℂ)
20		plysubcl 26175	. . . . . . . 8 ⊢ (((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ) ∧ (ℂ × {1}) ∈ (Poly‘ℂ)) → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_f − (ℂ × {1})) ∈ (Poly‘ℂ))
21	16, 19, 20	sylancl 587	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_f − (ℂ × {1})) ∈ (Poly‘ℂ))
22	9, 21	eqeltrrd 2836	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ∈ (Poly‘ℂ))
23		0cn 11125	. . . . . . 7 ⊢ 0 ∈ ℂ
24		neg1ne0 12135	. . . . . . . 8 ⊢ -1 ≠ 0
25	13	0expd 14090	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑(ϕ‘𝑁)) = 0)
26	25	oveq1d 7371	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0↑(ϕ‘𝑁)) − 1) = (0 − 1))
27		oveq1 7363	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧↑(ϕ‘𝑁)) = (0↑(ϕ‘𝑁)))
28	27	oveq1d 7371	. . . . . . . . . . . 12 ⊢ (𝑧 = 0 → ((𝑧↑(ϕ‘𝑁)) − 1) = ((0↑(ϕ‘𝑁)) − 1))
29		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))
30		ovex 7389	. . . . . . . . . . . 12 ⊢ ((0↑(ϕ‘𝑁)) − 1) ∈ V
31	28, 29, 30	fvmpt 6936	. . . . . . . . . . 11 ⊢ (0 ∈ ℂ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) = ((0↑(ϕ‘𝑁)) − 1))
32	23, 31	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) = ((0↑(ϕ‘𝑁)) − 1)
33		df-neg 11369	. . . . . . . . . 10 ⊢ -1 = (0 − 1)
34	26, 32, 33	3eqtr4g 2795	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) = -1)
35	34	neeq1d 2989	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0 ↔ -1 ≠ 0))
36	24, 35	mpbiri 258	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0)
37		ne0p 26160	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0) → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
38	23, 36, 37	sylancr 588	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
39	29	mptiniseg 6192	. . . . . . . . 9 ⊢ (0 ∈ ℂ → (^◡(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) “ {0}) = {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})
40	23, 39	ax-mp 5	. . . . . . . 8 ⊢ (^◡(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) “ {0}) = {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}
41	40	eqcomi 2744	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} = (^◡(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) “ {0})
42	41	fta1 26262	. . . . . 6 ⊢ (((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ∈ (Poly‘ℂ) ∧ (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝) → ({𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ∈ Fin ∧ (♯‘{𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ≤ (deg‘(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)))))
43	22, 38, 42	syl2anc 585	. . . . 5 ⊢ (𝑁 ∈ ℕ → ({𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ∈ Fin ∧ (♯‘{𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ≤ (deg‘(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)))))
44	43	simpld 494	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ∈ Fin)
45		unfi 9094	. . . 4 ⊢ (({0} ∈ Fin ∧ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ∈ Fin) → ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ∈ Fin)
46	1, 44, 45	sylancr 588	. . 3 ⊢ (𝑁 ∈ ℕ → ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ∈ Fin)
47		eqid 2735	. . . 4 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
48		eqid 2735	. . . 4 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁))
49	47, 48	znfi 21528	. . 3 ⊢ (𝑁 ∈ ℕ → (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)
50		mapfi 9247	. . 3 ⊢ ((({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ∈ Fin ∧ (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin) → (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁))) ∈ Fin)
51	46, 49, 50	syl2anc 585	. 2 ⊢ (𝑁 ∈ ℕ → (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁))) ∈ Fin)
52		dchrabl.g	. . . . . . . 8 ⊢ 𝐺 = (DChr‘𝑁)
53		dchrfi.b	. . . . . . . 8 ⊢ 𝐷 = (Base‘𝐺)
54		simpr 484	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
55	52, 47, 53, 48, 54	dchrf 27193	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ)
56	55	ffnd 6658	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 Fn (Base‘(ℤ/nℤ‘𝑁)))
57		df-ne 2931	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0)
58		fvex 6842	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
59	58	elsn 4572	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ {0} ↔ (𝑓‘𝑥) = 0)
60	57, 59	xchbinxr 335	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) ∈ {0})
61		oveq1 7363	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑧↑(ϕ‘𝑁)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
62	61	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑓‘𝑥) → ((𝑧↑(ϕ‘𝑁)) − 1) = (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1))
63	62	eqeq1d 2737	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓‘𝑥) → (((𝑧↑(ϕ‘𝑁)) − 1) = 0 ↔ (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0))
64		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
65		ffvelcdm 7022	. . . . . . . . . . . . 13 ⊢ ((𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ℂ)
66	55, 64, 65	syl2an 597	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ∈ ℂ)
67	52, 47, 53	dchrmhm 27192	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld))
68		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑓 ∈ 𝐷)
69	67, 68	sselid 3915	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑓 ∈ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld)))
70	14	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ₀)
71		simprl 771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
72		eqid 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘(ℤ/nℤ‘𝑁)) = (mulGrp‘(ℤ/nℤ‘𝑁))
73	72, 48	mgpbas 20115	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(mulGrp‘(ℤ/nℤ‘𝑁)))
74		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(ℤ/nℤ‘𝑁))) = (._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
75		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
76	73, 74, 75	mhmmulg 19080	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld)) ∧ (ϕ‘𝑁) ∈ ℕ₀ ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘((ϕ‘𝑁)(._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))𝑥)) = ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)))
77	69, 70, 71, 76	syl3anc 1374	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘((ϕ‘𝑁)(._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))𝑥)) = ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)))
78		nnnn0 12433	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
79	47	zncrng 21513	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ CRing)
81		crngring 20215	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring)
82	80, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ Ring)
83	82	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ℤ/nℤ‘𝑁) ∈ Ring)
84		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Unit‘(ℤ/nℤ‘𝑁))
85		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) = ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))
86	84, 85	unitgrp 20352	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) ∈ Grp)
87	83, 86	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) ∈ Grp)
88	47, 84	znunithash 21533	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
89	88, 14	eqeltrd 2835	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀)
90		fvex 6842	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) ∈ V
91		hashclb 14309	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Unit‘(ℤ/nℤ‘𝑁)) ∈ V → ((Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin ↔ (♯‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀))
92	90, 91	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin ↔ (♯‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀)
93	89, 92	sylibr 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
94	93	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
95		simprr 773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ≠ 0)
96	52, 47, 53, 48, 84, 68, 71	dchrn0 27201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((𝑓‘𝑥) ≠ 0 ↔ 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁))))
97	95, 96	mpbid 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁)))
98	84, 85	unitgrpbas 20351	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Base‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
99		eqid 2735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
100	98, 99	oddvds2 19530	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) ∈ Grp ∧ (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin ∧ 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁))) → ((od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))‘𝑥) ∥ (♯‘(Unit‘(ℤ/nℤ‘𝑁))))
101	87, 94, 97, 100	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))‘𝑥) ∥ (♯‘(Unit‘(ℤ/nℤ‘𝑁))))
102	88	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
103	101, 102	breqtrd 5100	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))‘𝑥) ∥ (ϕ‘𝑁))
104	13	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ)
105	104	nnzd 12539	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℤ)
106		eqid 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
107		eqid 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
108	98, 99, 106, 107	oddvds 19511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) ∈ Grp ∧ 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁)) ∧ (ϕ‘𝑁) ∈ ℤ) → (((od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))‘𝑥) ∥ (ϕ‘𝑁) ↔ ((ϕ‘𝑁)(._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))𝑥) = (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))))
109	87, 97, 105, 108	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))‘𝑥) ∥ (ϕ‘𝑁) ↔ ((ϕ‘𝑁)(._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))𝑥) = (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))))
110	103, 109	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((ϕ‘𝑁)(._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))𝑥) = (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))))
111	84, 72	unitsubm 20355	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (Unit‘(ℤ/nℤ‘𝑁)) ∈ (SubMnd‘(mulGrp‘(ℤ/nℤ‘𝑁))))
112	83, 111	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (Unit‘(ℤ/nℤ‘𝑁)) ∈ (SubMnd‘(mulGrp‘(ℤ/nℤ‘𝑁))))
113	74, 85, 106	submmulg 19083	. . . . . . . . . . . . . . . . . . 19 ⊢ (((Unit‘(ℤ/nℤ‘𝑁)) ∈ (SubMnd‘(mulGrp‘(ℤ/nℤ‘𝑁))) ∧ (ϕ‘𝑁) ∈ ℕ₀ ∧ 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁))) → ((ϕ‘𝑁)(._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))𝑥) = ((ϕ‘𝑁)(._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))𝑥))
114	112, 70, 97, 113	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((ϕ‘𝑁)(._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))𝑥) = ((ϕ‘𝑁)(._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))𝑥))
115		eqid 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(ℤ/nℤ‘𝑁))
116	72, 115	ringidval 20153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (0_g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
117	85, 116	subm0 18772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Unit‘(ℤ/nℤ‘𝑁)) ∈ (SubMnd‘(mulGrp‘(ℤ/nℤ‘𝑁))) → (1_r‘(ℤ/nℤ‘𝑁)) = (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))))
118	112, 117	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (1_r‘(ℤ/nℤ‘𝑁)) = (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))))
119	110, 114, 118	3eqtr4d 2780	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((ϕ‘𝑁)(._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))𝑥) = (1_r‘(ℤ/nℤ‘𝑁)))
120	119	fveq2d 6833	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘((ϕ‘𝑁)(._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))𝑥)) = (𝑓‘(1_r‘(ℤ/nℤ‘𝑁))))
121	77, 120	eqtr3d 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)) = (𝑓‘(1_r‘(ℤ/nℤ‘𝑁))))
122		cnfldexp 21374	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
123	66, 70, 122	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
124		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
125		cnfld1 21366	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1_r‘ℂ_fld)
126	124, 125	ringidval 20153	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
127	116, 126	mhm0 18751	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld)) → (𝑓‘(1_r‘(ℤ/nℤ‘𝑁))) = 1)
128	69, 127	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘(1_r‘(ℤ/nℤ‘𝑁))) = 1)
129	121, 123, 128	3eqtr3d 2778	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((𝑓‘𝑥)↑(ϕ‘𝑁)) = 1)
130	129	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = (1 − 1))
131		1m1e0 12242	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
132	130, 131	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0)
133	63, 66, 132	elrabd 3633	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})
134	133	expr 456	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → ((𝑓‘𝑥) ≠ 0 → (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
135	60, 134	biimtrrid 243	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (¬ (𝑓‘𝑥) ∈ {0} → (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
136	135	orrd 864	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → ((𝑓‘𝑥) ∈ {0} ∨ (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
137		elun 4085	. . . . . . . 8 ⊢ ((𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↔ ((𝑓‘𝑥) ∈ {0} ∨ (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
138	136, 137	sylibr 234	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
139	138	ralrimiva 3127	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → ∀𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
140		ffnfv 7060	. . . . . 6 ⊢ (𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↔ (𝑓 Fn (Base‘(ℤ/nℤ‘𝑁)) ∧ ∀𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})))
141	56, 139, 140	sylanbrc 584	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
142	141	ex 412	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})))
143	46, 49	elmapd 8776	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁))) ↔ 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})))
144	142, 143	sylibrd 259	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓 ∈ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁)))))
145	144	ssrdv 3923	. 2 ⊢ (𝑁 ∈ ℕ → 𝐷 ⊆ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁))))
146	51, 145	ssfid 9168	1 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 ∀wral 3049 {crab 3387 Vcvv 3427 ∪ cun 3883 ⊆ wss 3885 {csn 4557 class class class wbr 5074 ↦ cmpt 5155 × cxp 5618 ^◡ccnv 5619 “ cima 5623 Fn wfn 6482 ⟶wf 6483 ‘cfv 6487 (class class class)co 7356 ∘_f cof 7618 ↑_m cmap 8762 Fincfn 8882 ℂcc 11025 0cc0 11027 1c1 11028 ≤ cle 11169 − cmin 11366 -cneg 11367 ℕcn 12163 ℕ₀cn0 12426 ℤcz 12513 ↑cexp 14012 ♯chash 14281 ∥ cdvds 16210 ϕcphi 16723 Basecbs 17168 ↾_s cress 17189 0_gc0g 17391 MndHom cmhm 18738 SubMndcsubmnd 18739 Grpcgrp 18898 ._gcmg 19032 odcod 19488 mulGrpcmgp 20110 1_rcur 20151 Ringcrg 20203 CRingccrg 20204 Unitcui 20324 ℂ_fldccnfld 21341 ℤ/nℤczn 21471 0_𝑝c0p 25624 Polycply 26137 degcdgr 26140 DChrcdchr 27183

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-disj 5042 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-oadd 8398 df-omul 8399 df-er 8632 df-ec 8634 df-qs 8638 df-map 8764 df-pm 8765 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9344 df-inf 9345 df-oi 9414 df-dju 9814 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-xnn0 12500 df-z 12514 df-dec 12634 df-uz 12778 df-rp 12932 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-hash 14282 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15439 df-rlim 15440 df-sum 15638 df-dvds 16211 df-gcd 16453 df-phi 16725 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-0g 17393 df-imas 17461 df-qus 17462 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-nsg 19089 df-eqg 19090 df-ghm 19177 df-od 19492 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-oppr 20306 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-rhm 20441 df-subrng 20512 df-subrg 20536 df-lmod 20846 df-lss 20916 df-lsp 20956 df-sra 21157 df-rgmod 21158 df-lidl 21195 df-rsp 21196 df-2idl 21237 df-cnfld 21342 df-zring 21416 df-zrh 21472 df-zn 21475 df-0p 25625 df-ply 26141 df-idp 26142 df-coe 26143 df-dgr 26144 df-quot 26245 df-dchr 27184

This theorem is referenced by: sumdchr2 27221 dchrhash 27222 rpvmasum2 27463 dchrisum0re 27464

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