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Theorem dchrfi 25025

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 8079	. . . 4 ⊢ {0} ∈ Fin
2		cnex 10055	. . . . . . . . 9 ⊢ ℂ ∈ V
3	2	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ℂ ∈ V)
4		ovexd 6720	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → (𝑧↑(ϕ‘𝑁)) ∈ V)
5		1cnd 10094	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
6		eqidd 2652	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) = (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))))
7		fconstmpt 5197	. . . . . . . . 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 6956	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_𝑓 − (ℂ × {1})) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)))
10		ssid 3657	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ℂ ⊆ ℂ)
12		1cnd 10094	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
13		phicl 15521	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
14	13	nnnn0d 11389	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ₀)
15		plypow 24006	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
16	11, 12, 14, 15	syl3anc 1366	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∈ (Poly‘ℂ))
17		ax-1cn 10032	. . . . . . . . 9 ⊢ 1 ∈ ℂ
18		plyconst 24007	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → (ℂ × {1}) ∈ (Poly‘ℂ))
19	10, 17, 18	mp2an 708	. . . . . . . 8 ⊢ (ℂ × {1}) ∈ (Poly‘ℂ)
20		plysubcl 24023	. . . . . . . 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 695	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ (𝑧↑(ϕ‘𝑁))) ∘_𝑓 − (ℂ × {1})) ∈ (Poly‘ℂ))
22	9, 21	eqeltrrd 2731	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ∈ (Poly‘ℂ))
23		0cn 10070	. . . . . . 7 ⊢ 0 ∈ ℂ
24		neg1ne0 11164	. . . . . . . 8 ⊢ -1 ≠ 0
25	13	0expd 13064	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (0↑(ϕ‘𝑁)) = 0)
26	25	oveq1d 6705	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ((0↑(ϕ‘𝑁)) − 1) = (0 − 1))
27		oveq1 6697	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧↑(ϕ‘𝑁)) = (0↑(ϕ‘𝑁)))
28	27	oveq1d 6705	. . . . . . . . . . . 12 ⊢ (𝑧 = 0 → ((𝑧↑(ϕ‘𝑁)) − 1) = ((0↑(ϕ‘𝑁)) − 1))
29		eqid 2651	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) = (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))
30		ovex 6718	. . . . . . . . . . . 12 ⊢ ((0↑(ϕ‘𝑁)) − 1) ∈ V
31	28, 29, 30	fvmpt 6321	. . . . . . . . . . 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 10307	. . . . . . . . . 10 ⊢ -1 = (0 − 1)
34	26, 32, 33	3eqtr4g 2710	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) = -1)
35	34	neeq1d 2882	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0 ↔ -1 ≠ 0))
36	24, 35	mpbiri 248	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0)
37		ne0p 24008	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ ((𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1))‘0) ≠ 0) → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
38	23, 36, 37	sylancr 696	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) ≠ 0_𝑝)
39	29	mptiniseg 5667	. . . . . . . . 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 2660	. . . . . . 7 ⊢ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} = (^◡(𝑧 ∈ ℂ ↦ ((𝑧↑(ϕ‘𝑁)) − 1)) “ {0})
42	41	fta1 24108	. . . . . 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 694	. . . . 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 474	. . . 4 ⊢ (𝑁 ∈ ℕ → {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ∈ Fin)
45		unfi 8268	. . . 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 696	. . 3 ⊢ (𝑁 ∈ ℕ → ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ∈ Fin)
47		eqid 2651	. . . 4 ⊢ (ℤ/nℤ‘𝑁) = (ℤ/nℤ‘𝑁)
48		eqid 2651	. . . 4 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(ℤ/nℤ‘𝑁))
49	47, 48	znfi 19956	. . 3 ⊢ (𝑁 ∈ ℕ → (Base‘(ℤ/nℤ‘𝑁)) ∈ Fin)
50		mapfi 8303	. . 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 694	. 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 476	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 ∈ 𝐷)
55	52, 47, 53, 48, 54	dchrf 25012	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ)
56		ffn 6083	. . . . . . 7 ⊢ (𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ → 𝑓 Fn (Base‘(ℤ/nℤ‘𝑁)))
57	55, 56	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓 Fn (Base‘(ℤ/nℤ‘𝑁)))
58		df-ne 2824	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0)
59		fvex 6239	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
60	59	elsn 4225	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ {0} ↔ (𝑓‘𝑥) = 0)
61	58, 60	xchbinxr 324	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) ∈ {0})
62		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
63		ffvelrn 6397	. . . . . . . . . . . . 13 ⊢ ((𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶ℂ ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ℂ)
64	55, 62, 63	syl2an 493	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ∈ ℂ)
65	52, 47, 53	dchrmhm 25011	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 ⊆ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld))
66		simplr 807	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑓 ∈ 𝐷)
67	65, 66	sseldi 3634	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑓 ∈ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld)))
68	14	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ₀)
69		simprl 809	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)))
70		eqid 2651	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘(ℤ/nℤ‘𝑁)) = (mulGrp‘(ℤ/nℤ‘𝑁))
71	70, 48	mgpbas 18541	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(ℤ/nℤ‘𝑁)) = (Base‘(mulGrp‘(ℤ/nℤ‘𝑁)))
72		eqid 2651	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘(ℤ/nℤ‘𝑁))) = (._g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
73		eqid 2651	. . . . . . . . . . . . . . . . . 18 ⊢ (._g‘(mulGrp‘ℂ_fld)) = (._g‘(mulGrp‘ℂ_fld))
74	71, 72, 73	mhmmulg 17630	. . . . . . . . . . . . . . . . 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)))
75	67, 68, 69, 74	syl3anc 1366	. . . . . . . . . . . . . . . 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)))
76		nnnn0 11337	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
77	47	zncrng 19941	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → (ℤ/nℤ‘𝑁) ∈ CRing)
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ CRing)
79		crngring 18604	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℤ/nℤ‘𝑁) ∈ CRing → (ℤ/nℤ‘𝑁) ∈ Ring)
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (ℤ/nℤ‘𝑁) ∈ Ring)
81	80	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ℤ/nℤ‘𝑁) ∈ Ring)
82		eqid 2651	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Unit‘(ℤ/nℤ‘𝑁))
83		eqid 2651	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) = ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))
84	82, 83	unitgrp 18713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → ((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))) ∈ Grp)
85	81, 84	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)
86	47, 82	znunithash 19961	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (#‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
87	86, 14	eqeltrd 2730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (#‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀)
88		fvex 6239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) ∈ V
89		hashclb 13187	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Unit‘(ℤ/nℤ‘𝑁)) ∈ V → ((Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin ↔ (#‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀))
90	88, 89	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin ↔ (#‘(Unit‘(ℤ/nℤ‘𝑁))) ∈ ℕ₀)
91	87, 90	sylibr 224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
92	91	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (Unit‘(ℤ/nℤ‘𝑁)) ∈ Fin)
93		simprr 811	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ≠ 0)
94	52, 47, 53, 48, 82, 66, 69	dchrn0 25020	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((𝑓‘𝑥) ≠ 0 ↔ 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁))))
95	93, 94	mpbid 222	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → 𝑥 ∈ (Unit‘(ℤ/nℤ‘𝑁)))
96	82, 83	unitgrpbas 18712	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Unit‘(ℤ/nℤ‘𝑁)) = (Base‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
97		eqid 2651	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (od‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
98	96, 97	oddvds2 18029	. . . . . . . . . . . . . . . . . . . . 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))))
99	85, 92, 95, 98	syl3anc 1366	. . . . . . . . . . . . . . . . . . . 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))))
100	86	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (#‘(Unit‘(ℤ/nℤ‘𝑁))) = (ϕ‘𝑁))
101	99, 100	breqtrd 4711	. . . . . . . . . . . . . . . . . . 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))
102	13	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℕ)
103	102	nnzd 11519	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (ϕ‘𝑁) ∈ ℤ)
104		eqid 2651	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (._g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
105		eqid 2651	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁)))) = (0_g‘((mulGrp‘(ℤ/nℤ‘𝑁)) ↾_s (Unit‘(ℤ/nℤ‘𝑁))))
106	96, 97, 104, 105	oddvds 18012	. . . . . . . . . . . . . . . . . . . 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))))))
107	85, 95, 103, 106	syl3anc 1366	. . . . . . . . . . . . . . . . . . 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))))))
108	101, 107	mpbid 222	. . . . . . . . . . . . . . . . . 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)))))
109	82, 70	unitsubm 18716	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ/nℤ‘𝑁) ∈ Ring → (Unit‘(ℤ/nℤ‘𝑁)) ∈ (SubMnd‘(mulGrp‘(ℤ/nℤ‘𝑁))))
110	81, 109	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (Unit‘(ℤ/nℤ‘𝑁)) ∈ (SubMnd‘(mulGrp‘(ℤ/nℤ‘𝑁))))
111	72, 83, 104	submmulg 17633	. . . . . . . . . . . . . . . . . . 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))
112	110, 68, 95, 111	syl3anc 1366	. . . . . . . . . . . . . . . . . 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))
113		eqid 2651	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (1_r‘(ℤ/nℤ‘𝑁))
114	70, 113	ringidval 18549	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘(ℤ/nℤ‘𝑁)) = (0_g‘(mulGrp‘(ℤ/nℤ‘𝑁)))
115	83, 114	subm0 17403	. . . . . . . . . . . . . . . . . . 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)))))
116	110, 115	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)))))
117	108, 112, 116	3eqtr4d 2695	. . . . . . . . . . . . . . . . 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)))
118	117	fveq2d 6233	. . . . . . . . . . . . . . . 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))))
119	75, 118	eqtr3d 2687	. . . . . . . . . . . . . . 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))))
120		cnfldexp 19827	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑥) ∈ ℂ ∧ (ϕ‘𝑁) ∈ ℕ₀) → ((ϕ‘𝑁)(._g‘(mulGrp‘ℂ_fld))(𝑓‘𝑥)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
121	64, 68, 120	syl2anc 694	. . . . . . . . . . . . . . 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)))
122		eqid 2651	. . . . . . . . . . . . . . . . . 18 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
123		cnfld1 19819	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (1_r‘ℂ_fld)
124	122, 123	ringidval 18549	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
125	114, 124	mhm0 17390	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ((mulGrp‘(ℤ/nℤ‘𝑁)) MndHom (mulGrp‘ℂ_fld)) → (𝑓‘(1_r‘(ℤ/nℤ‘𝑁))) = 1)
126	67, 125	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘(1_r‘(ℤ/nℤ‘𝑁))) = 1)
127	119, 121, 126	3eqtr3d 2693	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → ((𝑓‘𝑥)↑(ϕ‘𝑁)) = 1)
128	127	oveq1d 6705	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = (1 − 1))
129		1m1e0 11127	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
130	128, 129	syl6eq 2701	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0)
131		oveq1 6697	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑓‘𝑥) → (𝑧↑(ϕ‘𝑁)) = ((𝑓‘𝑥)↑(ϕ‘𝑁)))
132	131	oveq1d 6705	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑓‘𝑥) → ((𝑧↑(ϕ‘𝑁)) − 1) = (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1))
133	132	eqeq1d 2653	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑓‘𝑥) → (((𝑧↑(ϕ‘𝑁)) − 1) = 0 ↔ (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0))
134	133	elrab 3396	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0} ↔ ((𝑓‘𝑥) ∈ ℂ ∧ (((𝑓‘𝑥)↑(ϕ‘𝑁)) − 1) = 0))
135	64, 130, 134	sylanbrc 699	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ (𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁)) ∧ (𝑓‘𝑥) ≠ 0)) → (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})
136	135	expr 642	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → ((𝑓‘𝑥) ≠ 0 → (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
137	61, 136	syl5bir 233	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (¬ (𝑓‘𝑥) ∈ {0} → (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
138	137	orrd 392	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → ((𝑓‘𝑥) ∈ {0} ∨ (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
139		elun 3786	. . . . . . . 8 ⊢ ((𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↔ ((𝑓‘𝑥) ∈ {0} ∨ (𝑓‘𝑥) ∈ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
140	138, 139	sylibr 224	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) ∧ 𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))) → (𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
141	140	ralrimiva 2995	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → ∀𝑥 ∈ (Base‘(ℤ/nℤ‘𝑁))(𝑓‘𝑥) ∈ ({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
142		ffnfv 6428	. . . . . 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})))
143	57, 141, 142	sylanbrc 699	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑓 ∈ 𝐷) → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}))
144	143	ex 449	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓:(Base‘(ℤ/nℤ‘𝑁))⟶({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0})))
145	46, 49	elmapd 7913	. . . 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})))
146	144, 145	sylibrd 249	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑓 ∈ 𝐷 → 𝑓 ∈ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_𝑚 (Base‘(ℤ/nℤ‘𝑁)))))
147	146	ssrdv 3642	. 2 ⊢ (𝑁 ∈ ℕ → 𝐷 ⊆ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_𝑚 (Base‘(ℤ/nℤ‘𝑁))))
148		ssfi 8221	. 2 ⊢ (((({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_𝑚 (Base‘(ℤ/nℤ‘𝑁))) ∈ Fin ∧ 𝐷 ⊆ (({0} ∪ {𝑧 ∈ ℂ ∣ ((𝑧↑(ϕ‘𝑁)) − 1) = 0}) ↑_𝑚 (Base‘(ℤ/nℤ‘𝑁)))) → 𝐷 ∈ Fin)
149	51, 147, 148	syl2anc 694	1 ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 {crab 2945 Vcvv 3231 ∪ cun 3605 ⊆ wss 3607 {csn 4210 class class class wbr 4685 ↦ cmpt 4762 × cxp 5141 ^◡ccnv 5142 “ cima 5146 Fn wfn 5921 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ∘_𝑓 cof 6937 ↑_𝑚 cmap 7899 Fincfn 7997 ℂcc 9972 0cc0 9974 1c1 9975 ≤ cle 10113 − cmin 10304 -cneg 10305 ℕcn 11058 ℕ₀cn0 11330 ℤcz 11415 ↑cexp 12900 #chash 13157 ∥ cdvds 15027 ϕcphi 15516 Basecbs 15904 ↾_s cress 15905 0_gc0g 16147 MndHom cmhm 17380 SubMndcsubmnd 17381 Grpcgrp 17469 ._gcmg 17587 odcod 17990 mulGrpcmgp 18535 1_rcur 18547 Ringcrg 18593 CRingccrg 18594 Unitcui 18685 ℂ_fldccnfld 19794 ℤ/nℤczn 19899 0_𝑝c0p 23481 Polycply 23985 degcdgr 23988 DChrcdchr 25002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-disj 4653 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-omul 7610 df-er 7787 df-ec 7789 df-qs 7793 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-xnn0 11402 df-z 11416 df-dec 11532 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-rlim 14264 df-sum 14461 df-dvds 15028 df-gcd 15264 df-phi 15518 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-0g 16149 df-imas 16215 df-qus 16216 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-nsg 17639 df-eqg 17640 df-ghm 17705 df-od 17994 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-rnghom 18763 df-subrg 18826 df-lmod 18913 df-lss 18981 df-lsp 19020 df-sra 19220 df-rgmod 19221 df-lidl 19222 df-rsp 19223 df-2idl 19280 df-cnfld 19795 df-zring 19867 df-zrh 19900 df-zn 19903 df-0p 23482 df-ply 23989 df-idp 23990 df-coe 23991 df-dgr 23992 df-quot 24091 df-dchr 25003

This theorem is referenced by: sumdchr2 25040 dchrhash 25041 rpvmasum2 25246 dchrisum0re 25247

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