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Theorem dchrfi 26403

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 8834	. . . 4 ⊢ {0} ∈ Fin
2		cnex 10952	. . . . . . . . 9 ⊢ ℂ ∈ V
3	2	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ℂ ∈ V)
4		ovexd 7310	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → (𝑧↑(ϕ‘𝑁)) ∈ V)
5		1cnd 10970	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
6		eqidd 2739	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) = (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))))
7		fconstmpt 5649	. . . . . . . . 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 7553	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_f − (ℂ × {1})) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)))
10		ssid 3943	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ℂ ⊆ ℂ)
12		1cnd 10970	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
13		phicl 16470	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
14	13	nnnn0d 12293	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ₀)
15		plypow 25366	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
16	11, 12, 14, 15	syl3anc 1370	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
17		ax-1cn 10929	. . . . . . . . 9 ⊢ 1 ∈ ℂ
18		plyconst 25367	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → (ℂ × {1}) ∈ (Poly‘ℂ))
19	10, 17, 18	mp2an 689	. . . . . . . 8 ⊢ (ℂ × {1}) ∈ (Poly‘ℂ)
20		plysubcl 25383	. . . . . . . 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 586	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_f − (ℂ × {1})) ∈ (Poly‘ℂ))
22	9, 21	eqeltrrd 2840	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ∈ (Poly‘ℂ))
23		0cn 10967	. . . . . . 7 ⊢ 0 ∈ ℂ
24		neg1ne0 12089	. . . . . . . 8 ⊢ -1 ≠ 0
25	13	0expd 13857	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑(ϕ‘𝑁)) = 0)
26	25	oveq1d 7290	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0↑(ϕ‘𝑁)) − 1) = (0 − 1))
27		oveq1 7282	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧↑(ϕ‘𝑁)) = (0↑(ϕ‘𝑁)))
28	27	oveq1d 7290	. . . . . . . . . . . 12 ⊢ (𝑧 = 0 → ((𝑧↑(ϕ‘𝑁)) − 1) = ((0↑(ϕ‘𝑁)) − 1))
29		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))
30		ovex 7308	. . . . . . . . . . . 12 ⊢ ((0↑(ϕ‘𝑁)) − 1) ∈ V
31	28, 29, 30	fvmpt 6875	. . . . . . . . . . 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 11208	. . . . . . . . . 10 ⊢ -1 = (0 − 1)
34	26, 32, 33	3eqtr4g 2803	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) = -1)
35	34	neeq1d 3003	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0 ↔ -1 ≠ 0))
36	24, 35	mpbiri 257	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0)
37		ne0p 25368	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0) → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
38	23, 36, 37	sylancr 587	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
39	29	mptiniseg 6142	. . . . . . . . 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 2747	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} = (^◡(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) “ {0})
42	41	fta1 25468	. . . . . 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 584	. . . . 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 495	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ∈ Fin)
45		unfi 8955	. . . 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 587	. . 3 ⊢ (𝑁 ∈ ℕ → ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ∈ Fin)
47		eqid 2738	. . . 4 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
48		eqid 2738	. . . 4 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁))
49	47, 48	znfi 20767	. . 3 ⊢ (𝑁 ∈ ℕ → (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)
50		mapfi 9115	. . 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 584	. 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 485	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
55	52, 47, 53, 48, 54	dchrf 26390	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ)
56	55	ffnd 6601	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 Fn (Base‘(ℤ/nℤ‘𝑁)))
57		df-ne 2944	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0)
58		fvex 6787	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
59	58	elsn 4576	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ {0} ↔ (𝑓‘𝑥) = 0)
60	57, 59	xchbinxr 335	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) ∈ {0})
61		oveq1 7282	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑧↑(ϕ‘𝑁)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
62	61	oveq1d 7290	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑓‘𝑥) → ((𝑧↑(ϕ‘𝑁)) − 1) = (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1))
63	62	eqeq1d 2740	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓‘𝑥) → (((𝑧↑(ϕ‘𝑁)) − 1) = 0 ↔ (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0))
64		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
65		ffvelrn 6959	. . . . . . . . . . . . 13 ⊢ ((𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ℂ)
66	55, 64, 65	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ∈ ℂ)
67	52, 47, 53	dchrmhm 26389	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld))
68		simplr 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑓 ∈ 𝐷)
69	67, 68	sselid 3919	. . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ₀)
71		simprl 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
72		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘(ℤ/nℤ‘𝑁)) = (mulGrp‘(ℤ/nℤ‘𝑁))
73	72, 48	mgpbas 19726	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(mulGrp‘(ℤ/nℤ‘𝑁)))
74		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(ℤ/nℤ‘𝑁))) = (._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
75		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
76	73, 74, 75	mhmmulg 18744	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 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 12240	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
79	47	zncrng 20752	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ CRing)
81		crngring 19795	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring)
82	80, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ Ring)
83	82	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ℤ/nℤ‘𝑁) ∈ Ring)
84		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Unit‘(ℤ/nℤ‘𝑁))
85		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) = ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))
86	84, 85	unitgrp 19909	. . . . . . . . . . . . . . . . . . . . . 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 20772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
89	88, 14	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀)
90		fvex 6787	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) ∈ V
91		hashclb 14073	. . . . . . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
94	93	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
95		simprr 770	. . . . . . . . . . . . . . . . . . . . . 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 26398	. . . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁)))
98	84, 85	unitgrpbas 19908	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Base‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
99		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 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 19173	. . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ)
105	104	nnzd 12425	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℤ)
106		eqid 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
107		eqid 2738	. . . . . . . . . . . . . . . . . . . . 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 19155	. . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . 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 19912	. . . . . . . . . . . . . . . . . . . 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 18747	. . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(ℤ/nℤ‘𝑁))
116	72, 115	ringidval 19739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (0_g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
117	85, 116	subm0 18454	. . . . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 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 20631	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
123	66, 70, 122	syl2anc 584	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
125		cnfld1 20623	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1_r‘ℂ_fld)
126	124, 125	ringidval 19739	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
127	116, 126	mhm0 18438	. . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((𝑓‘𝑥)↑(ϕ‘𝑁)) = 1)
130	129	oveq1d 7290	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = (1 − 1))
131		1m1e0 12045	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
132	130, 131	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0)
133	63, 66, 132	elrabd 3626	. . . . . . . . . . 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 457	. . . . . . . . . 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	syl5bir 242	. . . . . . . . 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 860	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → ((𝑓‘𝑥) ∈ {0} ∨ (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
137		elun 4083	. . . . . . . 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 233	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
139	138	ralrimiva 3103	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → ∀𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
140		ffnfv 6992	. . . . . 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 583	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
142	141	ex 413	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})))
143	46, 49	elmapd 8629	. . . 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 258	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓 ∈ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁)))))
145	144	ssrdv 3927	. 2 ⊢ (𝑁 ∈ ℕ → 𝐷 ⊆ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁))))
146	51, 145	ssfid 9042	1 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 {crab 3068 Vcvv 3432 ∪ cun 3885 ⊆ wss 3887 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ^◡ccnv 5588 “ cima 5592 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ↑_m cmap 8615 Fincfn 8733 ℂcc 10869 0cc0 10871 1c1 10872 ≤ cle 11010 − cmin 11205 -cneg 11206 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ↑cexp 13782 ♯chash 14044 ∥ cdvds 15963 ϕcphi 16465 Basecbs 16912 ↾_s cress 16941 0_gc0g 17150 MndHom cmhm 18428 SubMndcsubmnd 18429 Grpcgrp 18577 ._gcmg 18700 odcod 19132 mulGrpcmgp 19720 1_rcur 19737 Ringcrg 19783 CRingccrg 19784 Unitcui 19881 ℂ_fldccnfld 20597 ℤ/nℤczn 20704 0_𝑝c0p 24833 Polycply 25345 degcdgr 25348 DChrcdchr 26380

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-omul 8302 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-dvds 15964 df-gcd 16202 df-phi 16467 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-imas 17219 df-qus 17220 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-nsg 18753 df-eqg 18754 df-ghm 18832 df-od 19136 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-rnghom 19959 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-sra 20434 df-rgmod 20435 df-lidl 20436 df-rsp 20437 df-2idl 20503 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-zn 20708 df-0p 24834 df-ply 25349 df-idp 25350 df-coe 25351 df-dgr 25352 df-quot 25451 df-dchr 26381

This theorem is referenced by: sumdchr2 26418 dchrhash 26419 rpvmasum2 26660 dchrisum0re 26661

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