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 |
|
dchrzrh1.a |
⊢ ( 𝜑 → 𝐴 ∈ ℤ ) |
7 |
|
dchrzrh1.c |
⊢ ( 𝜑 → 𝐶 ∈ ℤ ) |
8 |
1 3
|
dchrrcl |
⊢ ( 𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ ) |
9 |
5 8
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
10 |
9
|
nnnn0d |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
11 |
2
|
zncrng |
⊢ ( 𝑁 ∈ ℕ0 → 𝑍 ∈ CRing ) |
12 |
10 11
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ CRing ) |
13 |
|
crngring |
⊢ ( 𝑍 ∈ CRing → 𝑍 ∈ Ring ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ Ring ) |
15 |
4
|
zrhrhm |
⊢ ( 𝑍 ∈ Ring → 𝐿 ∈ ( ℤring RingHom 𝑍 ) ) |
16 |
14 15
|
syl |
⊢ ( 𝜑 → 𝐿 ∈ ( ℤring RingHom 𝑍 ) ) |
17 |
|
zringbas |
⊢ ℤ = ( Base ‘ ℤring ) |
18 |
|
zringmulr |
⊢ · = ( .r ‘ ℤring ) |
19 |
|
eqid |
⊢ ( .r ‘ 𝑍 ) = ( .r ‘ 𝑍 ) |
20 |
17 18 19
|
rhmmul |
⊢ ( ( 𝐿 ∈ ( ℤring RingHom 𝑍 ) ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ) → ( 𝐿 ‘ ( 𝐴 · 𝐶 ) ) = ( ( 𝐿 ‘ 𝐴 ) ( .r ‘ 𝑍 ) ( 𝐿 ‘ 𝐶 ) ) ) |
21 |
16 6 7 20
|
syl3anc |
⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐴 · 𝐶 ) ) = ( ( 𝐿 ‘ 𝐴 ) ( .r ‘ 𝑍 ) ( 𝐿 ‘ 𝐶 ) ) ) |
22 |
21
|
fveq2d |
⊢ ( 𝜑 → ( 𝑋 ‘ ( 𝐿 ‘ ( 𝐴 · 𝐶 ) ) ) = ( 𝑋 ‘ ( ( 𝐿 ‘ 𝐴 ) ( .r ‘ 𝑍 ) ( 𝐿 ‘ 𝐶 ) ) ) ) |
23 |
1 2 3
|
dchrmhm |
⊢ 𝐷 ⊆ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) |
24 |
23 5
|
sselid |
⊢ ( 𝜑 → 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝑍 ) = ( Base ‘ 𝑍 ) |
26 |
17 25
|
rhmf |
⊢ ( 𝐿 ∈ ( ℤring RingHom 𝑍 ) → 𝐿 : ℤ ⟶ ( Base ‘ 𝑍 ) ) |
27 |
16 26
|
syl |
⊢ ( 𝜑 → 𝐿 : ℤ ⟶ ( Base ‘ 𝑍 ) ) |
28 |
27 6
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐴 ) ∈ ( Base ‘ 𝑍 ) ) |
29 |
27 7
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐶 ) ∈ ( Base ‘ 𝑍 ) ) |
30 |
|
eqid |
⊢ ( mulGrp ‘ 𝑍 ) = ( mulGrp ‘ 𝑍 ) |
31 |
30 25
|
mgpbas |
⊢ ( Base ‘ 𝑍 ) = ( Base ‘ ( mulGrp ‘ 𝑍 ) ) |
32 |
30 19
|
mgpplusg |
⊢ ( .r ‘ 𝑍 ) = ( +g ‘ ( mulGrp ‘ 𝑍 ) ) |
33 |
|
eqid |
⊢ ( mulGrp ‘ ℂfld ) = ( mulGrp ‘ ℂfld ) |
34 |
|
cnfldmul |
⊢ · = ( .r ‘ ℂfld ) |
35 |
33 34
|
mgpplusg |
⊢ · = ( +g ‘ ( mulGrp ‘ ℂfld ) ) |
36 |
31 32 35
|
mhmlin |
⊢ ( ( 𝑋 ∈ ( ( mulGrp ‘ 𝑍 ) MndHom ( mulGrp ‘ ℂfld ) ) ∧ ( 𝐿 ‘ 𝐴 ) ∈ ( Base ‘ 𝑍 ) ∧ ( 𝐿 ‘ 𝐶 ) ∈ ( Base ‘ 𝑍 ) ) → ( 𝑋 ‘ ( ( 𝐿 ‘ 𝐴 ) ( .r ‘ 𝑍 ) ( 𝐿 ‘ 𝐶 ) ) ) = ( ( 𝑋 ‘ ( 𝐿 ‘ 𝐴 ) ) · ( 𝑋 ‘ ( 𝐿 ‘ 𝐶 ) ) ) ) |
37 |
24 28 29 36
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 ‘ ( ( 𝐿 ‘ 𝐴 ) ( .r ‘ 𝑍 ) ( 𝐿 ‘ 𝐶 ) ) ) = ( ( 𝑋 ‘ ( 𝐿 ‘ 𝐴 ) ) · ( 𝑋 ‘ ( 𝐿 ‘ 𝐶 ) ) ) ) |
38 |
22 37
|
eqtrd |
⊢ ( 𝜑 → ( 𝑋 ‘ ( 𝐿 ‘ ( 𝐴 · 𝐶 ) ) ) = ( ( 𝑋 ‘ ( 𝐿 ‘ 𝐴 ) ) · ( 𝑋 ‘ ( 𝐿 ‘ 𝐶 ) ) ) ) |