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Theorem dchrfi 25533

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 8391	. . . 4 ⊢ {0} ∈ Fin
2		cnex 10416	. . . . . . . . 9 ⊢ ℂ ∈ V
3	2	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ℂ ∈ V)
4		ovexd 7010	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → (𝑧↑(ϕ‘𝑁)) ∈ V)
5		1cnd 10434	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
6		eqidd 2780	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) = (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))))
7		fconstmpt 5464	. . . . . . . . 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 7244	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_𝑓 − (ℂ × {1})) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)))
10		ssid 3880	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ℂ ⊆ ℂ)
12		1cnd 10434	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
13		phicl 15962	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
14	13	nnnn0d 11767	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ₀)
15		plypow 24498	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
16	11, 12, 14, 15	syl3anc 1351	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
17		ax-1cn 10393	. . . . . . . . 9 ⊢ 1 ∈ ℂ
18		plyconst 24499	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → (ℂ × {1}) ∈ (Poly‘ℂ))
19	10, 17, 18	mp2an 679	. . . . . . . 8 ⊢ (ℂ × {1}) ∈ (Poly‘ℂ)
20		plysubcl 24515	. . . . . . . 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 577	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_𝑓 − (ℂ × {1})) ∈ (Poly‘ℂ))
22	9, 21	eqeltrrd 2868	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ∈ (Poly‘ℂ))
23		0cn 10431	. . . . . . 7 ⊢ 0 ∈ ℂ
24		neg1ne0 11563	. . . . . . . 8 ⊢ -1 ≠ 0
25	13	0expd 13318	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑(ϕ‘𝑁)) = 0)
26	25	oveq1d 6991	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0↑(ϕ‘𝑁)) − 1) = (0 − 1))
27		oveq1 6983	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧↑(ϕ‘𝑁)) = (0↑(ϕ‘𝑁)))
28	27	oveq1d 6991	. . . . . . . . . . . 12 ⊢ (𝑧 = 0 → ((𝑧↑(ϕ‘𝑁)) − 1) = ((0↑(ϕ‘𝑁)) − 1))
29		eqid 2779	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))
30		ovex 7008	. . . . . . . . . . . 12 ⊢ ((0↑(ϕ‘𝑁)) − 1) ∈ V
31	28, 29, 30	fvmpt 6595	. . . . . . . . . . 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 10673	. . . . . . . . . 10 ⊢ -1 = (0 − 1)
34	26, 32, 33	3eqtr4g 2840	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) = -1)
35	34	neeq1d 3027	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0 ↔ -1 ≠ 0))
36	24, 35	mpbiri 250	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0)
37		ne0p 24500	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0) → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
38	23, 36, 37	sylancr 578	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
39	29	mptiniseg 5932	. . . . . . . . 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 2788	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} = (^◡(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) “ {0})
42	41	fta1 24600	. . . . . 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 576	. . . . 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 487	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ∈ Fin)
45		unfi 8580	. . . 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 578	. . 3 ⊢ (𝑁 ∈ ℕ → ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ∈ Fin)
47		eqid 2779	. . . 4 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
48		eqid 2779	. . . 4 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁))
49	47, 48	znfi 20408	. . 3 ⊢ (𝑁 ∈ ℕ → (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)
50		mapfi 8615	. . 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 576	. 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 477	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
55	52, 47, 53, 48, 54	dchrf 25520	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ)
56	55	ffnd 6345	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 Fn (Base‘(ℤ/nℤ‘𝑁)))
57		df-ne 2969	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0)
58		fvex 6512	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
59	58	elsn 4456	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ {0} ↔ (𝑓‘𝑥) = 0)
60	57, 59	xchbinxr 327	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) ∈ {0})
61		oveq1 6983	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑧↑(ϕ‘𝑁)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
62	61	oveq1d 6991	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑓‘𝑥) → ((𝑧↑(ϕ‘𝑁)) − 1) = (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1))
63	62	eqeq1d 2781	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓‘𝑥) → (((𝑧↑(ϕ‘𝑁)) − 1) = 0 ↔ (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0))
64		simpl 475	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
65		ffvelrn 6674	. . . . . . . . . . . . 13 ⊢ ((𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ℂ)
66	55, 64, 65	syl2an 586	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ∈ ℂ)
67	52, 47, 53	dchrmhm 25519	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld))
68		simplr 756	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑓 ∈ 𝐷)
69	67, 68	sseldi 3857	. . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ₀)
71		simprl 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
72		eqid 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘(ℤ/nℤ‘𝑁)) = (mulGrp‘(ℤ/nℤ‘𝑁))
73	72, 48	mgpbas 18968	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(mulGrp‘(ℤ/nℤ‘𝑁)))
74		eqid 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(ℤ/nℤ‘𝑁))) = (._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
75		eqid 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
76	73, 74, 75	mhmmulg 18052	. . . . . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . . . . 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 11715	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
79	47	zncrng 20393	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ CRing)
81		crngring 19031	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring)
82	80, 81	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ Ring)
83	82	ad2antrr 713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ℤ/nℤ‘𝑁) ∈ Ring)
84		eqid 2779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Unit‘(ℤ/nℤ‘𝑁))
85		eqid 2779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) = ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))
86	84, 85	unitgrp 19140	. . . . . . . . . . . . . . . . . . . . . 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 20413	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
89	88, 14	eqeltrd 2867	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀)
90		fvex 6512	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) ∈ V
91		hashclb 13534	. . . . . . . . . . . . . . . . . . . . . . . 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 226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
94	93	ad2antrr 713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
95		simprr 760	. . . . . . . . . . . . . . . . . . . . . 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 25528	. . . . . . . . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁)))
98	84, 85	unitgrpbas 19139	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Base‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
99		eqid 2779	. . . . . . . . . . . . . . . . . . . . . 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 18454	. . . . . . . . . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (♯‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
103	101, 102	breqtrd 4955	. . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ)
105	104	nnzd 11899	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℤ)
106		eqid 2779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
107		eqid 2779	. . . . . . . . . . . . . . . . . . . . 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 18437	. . . . . . . . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . . . . . 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 19143	. . . . . . . . . . . . . . . . . . . 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 18055	. . . . . . . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(ℤ/nℤ‘𝑁))
116	72, 115	ringidval 18976	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (0_g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
117	85, 116	subm0 17824	. . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . 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 6503	. . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . 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 20280	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
123	66, 70, 122	syl2anc 576	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
125		cnfld1 20272	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1_r‘ℂ_fld)
126	124, 125	ringidval 18976	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
127	116, 126	mhm0 17811	. . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((𝑓‘𝑥)↑(ϕ‘𝑁)) = 1)
130	129	oveq1d 6991	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = (1 − 1))
131		1m1e0 11512	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
132	130, 131	syl6eq 2831	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0)
133	63, 66, 132	elrabd 3599	. . . . . . . . . . 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 449	. . . . . . . . . 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 235	. . . . . . . . 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 849	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → ((𝑓‘𝑥) ∈ {0} ∨ (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
137		elun 4015	. . . . . . . 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 226	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
139	138	ralrimiva 3133	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → ∀𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
140		ffnfv 6705	. . . . . 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 575	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
142	141	ex 405	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})))
143	46, 49	elmapd 8220	. . . 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 251	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓 ∈ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_𝑚 (Base‘(ℤ/nℤ‘𝑁)))))
145	144	ssrdv 3865	. 2 ⊢ (𝑁 ∈ ℕ → 𝐷 ⊆ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_𝑚 (Base‘(ℤ/nℤ‘𝑁))))
146	51, 145	ssfid 8536	1 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 ∀wral 3089 {crab 3093 Vcvv 3416 ∪ cun 3828 ⊆ wss 3830 {csn 4441 class class class wbr 4929 ↦ cmpt 5008 × cxp 5405 ^◡ccnv 5406 “ cima 5410 Fn wfn 6183 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ∘_𝑓 cof 7225 ↑_𝑚 cmap 8206 Fincfn 8306 ℂcc 10333 0cc0 10335 1c1 10336 ≤ cle 10475 − cmin 10670 -cneg 10671 ℕcn 11439 ℕ₀cn0 11707 ℤcz 11793 ↑cexp 13244 ♯chash 13505 ∥ cdvds 15467 ϕcphi 15957 Basecbs 16339 ↾_s cress 16340 0_gc0g 16569 MndHom cmhm 17801 SubMndcsubmnd 17802 Grpcgrp 17891 ._gcmg 18011 odcod 18414 mulGrpcmgp 18962 1_rcur 18974 Ringcrg 19020 CRingccrg 19021 Unitcui 19112 ℂ_fldccnfld 20247 ℤ/nℤczn 20352 0_𝑝c0p 23973 Polycply 24477 degcdgr 24480 DChrcdchr 25510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 ax-mulf 10415

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-disj 4898 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-omul 7910 df-er 8089 df-ec 8091 df-qs 8095 df-map 8208 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-oi 8769 df-dju 9124 df-card 9162 df-acn 9165 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-xnn0 11780 df-z 11794 df-dec 11912 df-uz 12059 df-rp 12205 df-fz 12709 df-fzo 12850 df-fl 12977 df-mod 13053 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-rlim 14707 df-sum 14904 df-dvds 15468 df-gcd 15704 df-phi 15959 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-starv 16436 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-0g 16571 df-imas 16637 df-qus 16638 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-nsg 18061 df-eqg 18062 df-ghm 18127 df-od 18418 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-cring 19023 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-rnghom 19190 df-subrg 19256 df-lmod 19358 df-lss 19426 df-lsp 19466 df-sra 19666 df-rgmod 19667 df-lidl 19668 df-rsp 19669 df-2idl 19726 df-cnfld 20248 df-zring 20320 df-zrh 20353 df-zn 20356 df-0p 23974 df-ply 24481 df-idp 24482 df-coe 24483 df-dgr 24484 df-quot 24583 df-dchr 25511

This theorem is referenced by: sumdchr2 25548 dchrhash 25549 rpvmasum2 25790 dchrisum0re 25791

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