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Metamath Proof Explorer

Theorem dchrn0

Description: A Dirichlet character is nonzero on the units of Z/nZ . (Contributed by Mario Carneiro, 18-Apr-2016)

Ref Expression
Hypotheses dchrmhm.g ⊢ 𝐺 = ( DChr ‘ 𝑁 )
dchrmhm.z ⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 )
dchrmhm.b ⊢ 𝐷 = ( Base ‘ 𝐺 )
dchrn0.b ⊢ 𝐵 = ( Base ‘ 𝑍 )
dchrn0.u ⊢ 𝑈 = ( Unit ‘ 𝑍 )
dchrn0.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐷 )
dchrn0.a ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
Assertion dchrn0 ( 𝜑 → ( ( 𝑋 ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ∈ 𝑈 ) )

Proof

Step Hyp Ref Expression
1 dchrmhm.g ⊢ 𝐺 = ( DChr ‘ 𝑁 )
2 dchrmhm.z ⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 )
3 dchrmhm.b ⊢ 𝐷 = ( Base ‘ 𝐺 )
4 dchrn0.b ⊢ 𝐵 = ( Base ‘ 𝑍 )
5 dchrn0.u ⊢ 𝑈 = ( Unit ‘ 𝑍 )
6 dchrn0.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐷 )
7 dchrn0.a ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
8 fveq2 ⊢ ( 𝑥 = 𝐴 → ( 𝑋 ‘ 𝑥 ) = ( 𝑋 ‘ 𝐴 ) )
9 8 neeq1d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑋 ‘ 𝑥 ) ≠ 0 ↔ ( 𝑋 ‘ 𝐴 ) ≠ 0 ) )
10 eleq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈 ) )
11 9 10 imbi12d ⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑋 ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ 𝑈 ) ↔ ( ( 𝑋 ‘ 𝐴 ) ≠ 0 → 𝐴 ∈ 𝑈 ) ) )
12 1 3 dchrrcl ⊢ ( 𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ )
13 6 12 syl ⊢ ( 𝜑 → 𝑁 ∈ ℕ )
14 1 2 4 5 13 3 dchrelbas2 ⊢ ( 𝜑 → ( 𝑋 ∈ 𝐷 ↔ ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑋 ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ 𝑈 ) ) ) )
15 6 14 mpbid ⊢ ( 𝜑 → ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ∧ ∀ 𝑥 ∈ 𝐵 ( ( 𝑋 ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ 𝑈 ) ) )
16 15 simprd ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( ( 𝑋 ‘ 𝑥 ) ≠ 0 → 𝑥 ∈ 𝑈 ) )
17 11 16 7 rspcdva ⊢ ( 𝜑 → ( ( 𝑋 ‘ 𝐴 ) ≠ 0 → 𝐴 ∈ 𝑈 ) )
18 17 imp ⊢ ( ( 𝜑 ∧ ( 𝑋 ‘ 𝐴 ) ≠ 0 ) → 𝐴 ∈ 𝑈 )
19 ax-1ne0 ⊢ 1 ≠ 0
20 19 a1i ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → 1 ≠ 0 )
21 13 nnnn0d ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
22 2 zncrng ⊢ ( 𝑁 ∈ ℕ0 → 𝑍 ∈ CRing )
23 crngring ⊢ ( 𝑍 ∈ CRing → 𝑍 ∈ Ring )
24 21 22 23 3syl ⊢ ( 𝜑 → 𝑍 ∈ Ring )
25 eqid ⊢ ( invr ‘ 𝑍 ) = ( invr ‘ 𝑍 )
26 eqid ⊢ ( .r ‘ 𝑍 ) = ( .r ‘ 𝑍 )
27 eqid ⊢ ( 1r ‘ 𝑍 ) = ( 1r ‘ 𝑍 )
28 5 25 26 27 unitrinv ⊢ ( ( 𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈 ) → ( 𝐴 ( .r ‘ 𝑍 ) ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) = ( 1r ‘ 𝑍 ) )
29 24 28 sylan ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( 𝐴 ( .r ‘ 𝑍 ) ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) = ( 1r ‘ 𝑍 ) )
30 29 fveq2d ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( 𝑋 ‘ ( 𝐴 ( .r ‘ 𝑍 ) ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) = ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) )
31 15 simpld ⊢ ( 𝜑 → 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) )
32 31 adantr ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) )
33 7 adantr ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → 𝐴 ∈ 𝐵 )
34 5 25 4 ringinvcl ⊢ ( ( 𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈 ) → ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ∈ 𝐵 )
35 24 34 sylan ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ∈ 𝐵 )
36 eqid ⊢ ( mulGrp ‘ 𝑍 ) = ( mulGrp ‘ 𝑍 )
37 36 4 mgpbas ⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑍 ) )
38 36 26 mgpplusg ⊢ ( .r ‘ 𝑍 ) = ( +g ‘ ( mulGrp ‘ 𝑍 ) )
39 eqid ⊢ ( mulGrp ‘ ℂfld ) = ( mulGrp ‘ ℂfld )
40 cnfldmul ⊢ · = ( .r ‘ ℂfld )
41 39 40 mgpplusg ⊢ · = ( +g ‘ ( mulGrp ‘ ℂfld ) )
42 37 38 41 mhmlin ⊢ ( ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ∧ 𝐴 ∈ 𝐵 ∧ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ∈ 𝐵 ) → ( 𝑋 ‘ ( 𝐴 ( .r ‘ 𝑍 ) ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) = ( ( 𝑋 ‘ 𝐴 ) · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) )
43 32 33 35 42 syl3anc ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( 𝑋 ‘ ( 𝐴 ( .r ‘ 𝑍 ) ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) = ( ( 𝑋 ‘ 𝐴 ) · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) )
44 36 27 ringidval ⊢ ( 1r ‘ 𝑍 ) = ( 0g ‘ ( mulGrp ‘ 𝑍 ) )
45 cnfld1 ⊢ 1 = ( 1r ‘ ℂfld )
46 39 45 ringidval ⊢ 1 = ( 0g ‘ ( mulGrp ‘ ℂfld ) )
47 44 46 mhm0 ⊢ ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) → ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) = 1 )
48 32 47 syl ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) = 1 )
49 30 43 48 3eqtr3d ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( ( 𝑋 ‘ 𝐴 ) · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) = 1 )
50 cnfldbas ⊢ ℂ = ( Base ‘ ℂfld )
51 39 50 mgpbas ⊢ ℂ = ( Base ‘ ( mulGrp ‘ ℂfld ) )
52 37 51 mhmf ⊢ ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) → 𝑋 : 𝐵 ⟶ ℂ )
53 32 52 syl ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → 𝑋 : 𝐵 ⟶ ℂ )
54 53 35 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ∈ ℂ )
55 54 mul02d ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( 0 · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) = 0 )
56 20 49 55 3netr4d ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( ( 𝑋 ‘ 𝐴 ) · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) ≠ ( 0 · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) )
57 oveq1 ⊢ ( ( 𝑋 ‘ 𝐴 ) = 0 → ( ( 𝑋 ‘ 𝐴 ) · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) = ( 0 · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) )
58 57 necon3i ⊢ ( ( ( 𝑋 ‘ 𝐴 ) · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) ≠ ( 0 · ( 𝑋 ‘ ( ( invr ‘ 𝑍 ) ‘ 𝐴 ) ) ) → ( 𝑋 ‘ 𝐴 ) ≠ 0 )
59 56 58 syl ⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ( 𝑋 ‘ 𝐴 ) ≠ 0 )
60 18 59 impbida ⊢ ( 𝜑 → ( ( 𝑋 ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ∈ 𝑈 ) )