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Theorem dchrfi 27582

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 9071	. . . 4 ⊢ {0} ∈ Fin
2		cnex 11281	. . . . . . . . 9 ⊢ ℂ ∈ V
3	2	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ℂ ∈ V)
4		ovexd 7455	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → (𝑧↑(ϕ‘𝑁)) ∈ V)
5		1cnd 11302	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
6		eqidd 2761	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) = (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))))
7		fconstmpt 5712	. . . . . . . . 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 7713	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_f − (ℂ × {1})) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)))
10		ssid 3930	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ℂ ⊆ ℂ)
12		1cnd 11302	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
13		phicl 16946	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
14	13	nnnn0d 12667	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ₀)
15		plypow 26523	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
16	11, 12, 14, 15	syl3anc 1398	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
17		ax-1cn 11258	. . . . . . . . 9 ⊢ 1 ∈ ℂ
18		plyconst 26524	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → (ℂ × {1}) ∈ (Poly‘ℂ))
19	10, 17, 18	mp2an 705	. . . . . . . 8 ⊢ (ℂ × {1}) ∈ (Poly‘ℂ)
20		plysubcl 26541	. . . . . . . 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 598	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_f − (ℂ × {1})) ∈ (Poly‘ℂ))
22	9, 21	eqeltrrd 2861	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ∈ (Poly‘ℂ))
23		0cn 11298	. . . . . . 7 ⊢ 0 ∈ ℂ
24		neg1ne0 12307	. . . . . . . 8 ⊢ -1 ≠ 0
25	13	0expd 14282	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑(ϕ‘𝑁)) = 0)
26	25	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0↑(ϕ‘𝑁)) − 1) = (0 − 1))
27		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧↑(ϕ‘𝑁)) = (0↑(ϕ‘𝑁)))
28	27	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (𝑧 = 0 → ((𝑧↑(ϕ‘𝑁)) − 1) = ((0↑(ϕ‘𝑁)) − 1))
29		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))
30		ovex 7453	. . . . . . . . . . . 12 ⊢ ((0↑(ϕ‘𝑁)) − 1) ∈ V
31	28, 29, 30	fvmpt 6993	. . . . . . . . . . 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 11544	. . . . . . . . . 10 ⊢ -1 = (0 − 1)
34	26, 32, 33	3eqtr4g 2820	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) = -1)
35	34	neeq1d 3014	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0 ↔ -1 ≠ 0))
36	24, 35	mpbiri 261	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0)
37		ne0p 26525	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0) → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
38	23, 36, 37	sylancr 599	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
39	29	mptiniseg 6240	. . . . . . . . 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 2769	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} = (^◡(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) “ {0})
42	41	fta1 26629	. . . . . 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 596	. . . . 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 500	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ∈ Fin)
45		unfi 9186	. . . 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 599	. . 3 ⊢ (𝑁 ∈ ℕ → ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ∈ Fin)
47		eqid 2760	. . . 4 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
48		eqid 2760	. . . 4 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁))
49	47, 48	znfi 21865	. . 3 ⊢ (𝑁 ∈ ℕ → (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)
50		mapfi 9337	. . 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 596	. 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 490	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
55	52, 47, 53, 48, 54	dchrf 27569	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ)
56	55	ffnd 6710	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 Fn (Base‘(ℤ/nℤ‘𝑁)))
57		df-ne 2956	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0)
58		fvex 6898	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
59	58	elsn 4598	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ {0} ↔ (𝑓‘𝑥) = 0)
60	57, 59	xchbinxr 338	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) ∈ {0})
61		oveq1 7427	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑧↑(ϕ‘𝑁)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
62	61	oveq1d 7435	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑓‘𝑥) → ((𝑧↑(ϕ‘𝑁)) − 1) = (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1))
63	62	eqeq1d 2762	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓‘𝑥) → (((𝑧↑(ϕ‘𝑁)) − 1) = 0 ↔ (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0))
64		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
65		ffvelcdm 7081	. . . . . . . . . . . . 13 ⊢ ((𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ℂ)
66	55, 64, 65	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ∈ ℂ)
67	52, 47, 53	dchrmhm 27568	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld))
68		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑓 ∈ 𝐷)
69	67, 68	sselid 3906	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ₀)
71		simprl 783	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
72		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘(ℤ/nℤ‘𝑁)) = (mulGrp‘(ℤ/nℤ‘𝑁))
73	72, 48	mgpbas 20365	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(mulGrp‘(ℤ/nℤ‘𝑁)))
74		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(ℤ/nℤ‘𝑁))) = (._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
75		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
76	73, 74, 75	mhmmulg 19325	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 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 12613	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
79	47	zncrng 21850	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
80	78, 79	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ CRing)
81		crngring 20472	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring)
82	80, 81	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ Ring)
83	82	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ℤ/nℤ‘𝑁) ∈ Ring)
84		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Unit‘(ℤ/nℤ‘𝑁))
85		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) = ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))
86	84, 85	unitgrp 20613	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) ∈ Grp)
87	83, 86	syl 18	. . . . . . . . . . . . . . . . . . . . 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 21870	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
89	88, 14	eqeltrd 2860	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀)
90		fvex 6898	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) ∈ V
91		hashclb 14502	. . . . . . . . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
94	93	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
95		simprr 785	. . . . . . . . . . . . . . . . . . . . . 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 27577	. . . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁)))
98	84, 85	unitgrpbas 20612	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Base‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
99		eqid 2760	. . . . . . . . . . . . . . . . . . . . . 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 19780	. . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
103	101, 102	breqtrd 5130	. . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ)
105	104	nnzd 12719	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℤ)
106		eqid 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
107		eqid 2760	. . . . . . . . . . . . . . . . . . . . 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 19761	. . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . 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 20616	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (Unit‘(ℤ/nℤ‘𝑁)) ∈ (SubMnd‘(mulGrp‘(ℤ/nℤ‘𝑁))))
112	83, 111	syl 18	. . . . . . . . . . . . . . . . . . 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 19328	. . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(ℤ/nℤ‘𝑁))
116	72, 115	ringidval 20409	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (0_g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
117	85, 116	subm0 19011	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . 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 21711	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
123	66, 70, 122	syl2anc 596	. . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
125		cnfld1 21703	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1_r‘ℂ_fld)
126	124, 125	ringidval 20409	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
127	116, 126	mhm0 18989	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld)) → (𝑓‘(1_r‘(ℤ/nℤ‘𝑁))) = 1)
128	69, 127	syl 18	. . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((𝑓‘𝑥)↑(ϕ‘𝑁)) = 1)
130	129	oveq1d 7435	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = (1 − 1))
131		1m1e0 12415	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
132	130, 131	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0)
133	63, 66, 132	elrabd 3646	. . . . . . . . . . 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 462	. . . . . . . . . 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 246	. . . . . . . . 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 877	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → ((𝑓‘𝑥) ∈ {0} ∨ (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
137		elun 4152	. . . . . . . 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 237	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
139	138	ralrimiva 3154	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → ∀𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
140		ffnfv 7119	. . . . . 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 595	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
142	141	ex 418	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})))
143	46, 49	elmapd 8860	. . . 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 262	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓 ∈ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁)))))
145	144	ssrdv 3914	. 2 ⊢ (𝑁 ∈ ℕ → 𝐷 ⊆ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_m (Base‘(ℤ/nℤ‘𝑁))))
146	51, 145	ssfid 9260	1 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 {crab 3412 Vcvv 3450 ⊆ wss 3895 ∪ cun 4149 {csn 4583 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 ^◡ccnv 5649 “ cima 5653 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∘_f cof 7691 ↑_m cmap 8847 Fincfn 8973 ℂcc 11198 0cc0 11200 1c1 11201 ≤ cle 11344 − cmin 11541 -cneg 11542 ℕcn 12335 ℕ₀cn0 12606 ℤcz 12693 ↑cexp 14204 ♯chash 14474 ∥ cdvds 16422 ϕcphi 16941 Basecbs 17387 ↾_s cress 17408 0_gc0g 17610 MndHom cmhm 18976 SubMndcsubmnd 18977 Grpcgrp 19144 ._gcmg 19277 odcod 19738 mulGrpcmgp 20360 1_rcur 20407 Ringcrg 20459 CRingccrg 20460 Unitcui 20585 ℂ_fldccnfld 21678 ℤ/nℤczn 21808 0_𝑝c0p 25990 Polycply 26502 degcdgr 26505 DChrcdchr 27559

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279 ax-mulf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-oadd 8480 df-omul 8481 df-er 8717 df-ec 8719 df-qs 8723 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-acn 10023 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-dvds 16423 df-gcd 16665 df-phi 16943 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-0g 17612 df-imas 17680 df-qus 17681 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-nsg 19334 df-eqg 19335 df-ghm 19428 df-od 19742 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-rhm 20702 df-subrng 20798 df-subrg 20822 df-lmod 21137 df-lss 21207 df-lsp 21247 df-sra 21448 df-rgmod 21449 df-lidl 21486 df-rsp 21487 df-2idl 21543 df-cnfld 21679 df-zring 21753 df-zrh 21809 df-zn 21812 df-0p 25991 df-ply 26506 df-idp 26507 df-coe 26508 df-dgr 26509 df-quot 26612 df-dchr 27560

This theorem is used by: sumdchr2 27597 dchrhash 27598 rpvmasum2 27839 dchrisum0re 27840

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