Step |
Hyp |
Ref |
Expression |
1 |
|
dchrmhm.g |
⊢ 𝐺 = ( DChr ‘ 𝑁 ) |
2 |
|
dchrmhm.z |
⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 ) |
3 |
|
dchrmhm.b |
⊢ 𝐷 = ( Base ‘ 𝐺 ) |
4 |
|
dchrelbas4.l |
⊢ 𝐿 = ( ℤRHom ‘ 𝑍 ) |
5 |
|
dchrzrh1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐷 ) |
6 |
1 3
|
dchrrcl |
⊢ ( 𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ ) |
7 |
5 6
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
8 |
7
|
nnnn0d |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
9 |
2
|
zncrng |
⊢ ( 𝑁 ∈ ℕ0 → 𝑍 ∈ CRing ) |
10 |
|
crngring |
⊢ ( 𝑍 ∈ CRing → 𝑍 ∈ Ring ) |
11 |
|
eqid |
⊢ ( 1r ‘ 𝑍 ) = ( 1r ‘ 𝑍 ) |
12 |
4 11
|
zrh1 |
⊢ ( 𝑍 ∈ Ring → ( 𝐿 ‘ 1 ) = ( 1r ‘ 𝑍 ) ) |
13 |
8 9 10 12
|
4syl |
⊢ ( 𝜑 → ( 𝐿 ‘ 1 ) = ( 1r ‘ 𝑍 ) ) |
14 |
13
|
fveq2d |
⊢ ( 𝜑 → ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) = ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) ) |
15 |
1 2 3
|
dchrmhm |
⊢ 𝐷 ⊆ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) |
16 |
15 5
|
sselid |
⊢ ( 𝜑 → 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ) |
17 |
|
eqid |
⊢ ( mulGrp ‘ 𝑍 ) = ( mulGrp ‘ 𝑍 ) |
18 |
17 11
|
ringidval |
⊢ ( 1r ‘ 𝑍 ) = ( 0g ‘ ( mulGrp ‘ 𝑍 ) ) |
19 |
|
eqid |
⊢ ( mulGrp ‘ ℂfld ) = ( mulGrp ‘ ℂfld ) |
20 |
|
cnfld1 |
⊢ 1 = ( 1r ‘ ℂfld ) |
21 |
19 20
|
ringidval |
⊢ 1 = ( 0g ‘ ( mulGrp ‘ ℂfld ) ) |
22 |
18 21
|
mhm0 |
⊢ ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) → ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) = 1 ) |
23 |
16 22
|
syl |
⊢ ( 𝜑 → ( 𝑋 ‘ ( 1r ‘ 𝑍 ) ) = 1 ) |
24 |
14 23
|
eqtrd |
⊢ ( 𝜑 → ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) = 1 ) |