Step |
Hyp |
Ref |
Expression |
1 |
|
dchrresb.g |
⊢ 𝐺 = ( DChr ‘ 𝑁 ) |
2 |
|
dchrresb.z |
⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 ) |
3 |
|
dchrresb.b |
⊢ 𝐷 = ( Base ‘ 𝐺 ) |
4 |
|
dchrresb.u |
⊢ 𝑈 = ( Unit ‘ 𝑍 ) |
5 |
|
dchrresb.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐷 ) |
6 |
|
dchrresb.Y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐷 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑍 ) = ( Base ‘ 𝑍 ) |
8 |
1 2 3 7 5
|
dchrf |
⊢ ( 𝜑 → 𝑋 : ( Base ‘ 𝑍 ) ⟶ ℂ ) |
9 |
8
|
ffnd |
⊢ ( 𝜑 → 𝑋 Fn ( Base ‘ 𝑍 ) ) |
10 |
1 2 3 7 6
|
dchrf |
⊢ ( 𝜑 → 𝑌 : ( Base ‘ 𝑍 ) ⟶ ℂ ) |
11 |
10
|
ffnd |
⊢ ( 𝜑 → 𝑌 Fn ( Base ‘ 𝑍 ) ) |
12 |
7 4
|
unitss |
⊢ 𝑈 ⊆ ( Base ‘ 𝑍 ) |
13 |
|
fvreseq |
⊢ ( ( ( 𝑋 Fn ( Base ‘ 𝑍 ) ∧ 𝑌 Fn ( Base ‘ 𝑍 ) ) ∧ 𝑈 ⊆ ( Base ‘ 𝑍 ) ) → ( ( 𝑋 ↾ 𝑈 ) = ( 𝑌 ↾ 𝑈 ) ↔ ∀ 𝑘 ∈ 𝑈 ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |
14 |
12 13
|
mpan2 |
⊢ ( ( 𝑋 Fn ( Base ‘ 𝑍 ) ∧ 𝑌 Fn ( Base ‘ 𝑍 ) ) → ( ( 𝑋 ↾ 𝑈 ) = ( 𝑌 ↾ 𝑈 ) ↔ ∀ 𝑘 ∈ 𝑈 ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |
15 |
9 11 14
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑋 ↾ 𝑈 ) = ( 𝑌 ↾ 𝑈 ) ↔ ∀ 𝑘 ∈ 𝑈 ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |
16 |
1 2 3 4 5 6
|
dchreq |
⊢ ( 𝜑 → ( 𝑋 = 𝑌 ↔ ∀ 𝑘 ∈ 𝑈 ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |
17 |
15 16
|
bitr4d |
⊢ ( 𝜑 → ( ( 𝑋 ↾ 𝑈 ) = ( 𝑌 ↾ 𝑈 ) ↔ 𝑋 = 𝑌 ) ) |