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 |
|
eldif |
⊢ ( 𝑘 ∈ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ↔ ( 𝑘 ∈ ( Base ‘ 𝑍 ) ∧ ¬ 𝑘 ∈ 𝑈 ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑍 ) = ( Base ‘ 𝑍 ) |
9 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → 𝑋 ∈ 𝐷 ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → 𝑘 ∈ ( Base ‘ 𝑍 ) ) |
11 |
1 2 3 8 4 9 10
|
dchrn0 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → ( ( 𝑋 ‘ 𝑘 ) ≠ 0 ↔ 𝑘 ∈ 𝑈 ) ) |
12 |
11
|
biimpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → ( ( 𝑋 ‘ 𝑘 ) ≠ 0 → 𝑘 ∈ 𝑈 ) ) |
13 |
12
|
necon1bd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → ( ¬ 𝑘 ∈ 𝑈 → ( 𝑋 ‘ 𝑘 ) = 0 ) ) |
14 |
13
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑍 ) ∧ ¬ 𝑘 ∈ 𝑈 ) ) → ( 𝑋 ‘ 𝑘 ) = 0 ) |
15 |
7 14
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ) → ( 𝑋 ‘ 𝑘 ) = 0 ) |
16 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → 𝑌 ∈ 𝐷 ) |
17 |
1 2 3 8 4 16 10
|
dchrn0 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → ( ( 𝑌 ‘ 𝑘 ) ≠ 0 ↔ 𝑘 ∈ 𝑈 ) ) |
18 |
17
|
biimpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → ( ( 𝑌 ‘ 𝑘 ) ≠ 0 → 𝑘 ∈ 𝑈 ) ) |
19 |
18
|
necon1bd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( Base ‘ 𝑍 ) ) → ( ¬ 𝑘 ∈ 𝑈 → ( 𝑌 ‘ 𝑘 ) = 0 ) ) |
20 |
19
|
impr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( Base ‘ 𝑍 ) ∧ ¬ 𝑘 ∈ 𝑈 ) ) → ( 𝑌 ‘ 𝑘 ) = 0 ) |
21 |
7 20
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ) → ( 𝑌 ‘ 𝑘 ) = 0 ) |
22 |
15 21
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ) → ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) |
23 |
22
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) |
24 |
1 2 3 8 5
|
dchrf |
⊢ ( 𝜑 → 𝑋 : ( Base ‘ 𝑍 ) ⟶ ℂ ) |
25 |
24
|
ffnd |
⊢ ( 𝜑 → 𝑋 Fn ( Base ‘ 𝑍 ) ) |
26 |
1 2 3 8 6
|
dchrf |
⊢ ( 𝜑 → 𝑌 : ( Base ‘ 𝑍 ) ⟶ ℂ ) |
27 |
26
|
ffnd |
⊢ ( 𝜑 → 𝑌 Fn ( Base ‘ 𝑍 ) ) |
28 |
|
eqfnfv |
⊢ ( ( 𝑋 Fn ( Base ‘ 𝑍 ) ∧ 𝑌 Fn ( Base ‘ 𝑍 ) ) → ( 𝑋 = 𝑌 ↔ ∀ 𝑘 ∈ ( Base ‘ 𝑍 ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |
29 |
25 27 28
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 = 𝑌 ↔ ∀ 𝑘 ∈ ( Base ‘ 𝑍 ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |
30 |
8 4
|
unitss |
⊢ 𝑈 ⊆ ( Base ‘ 𝑍 ) |
31 |
|
undif |
⊢ ( 𝑈 ⊆ ( Base ‘ 𝑍 ) ↔ ( 𝑈 ∪ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ) = ( Base ‘ 𝑍 ) ) |
32 |
30 31
|
mpbi |
⊢ ( 𝑈 ∪ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ) = ( Base ‘ 𝑍 ) |
33 |
32
|
raleqi |
⊢ ( ∀ 𝑘 ∈ ( 𝑈 ∪ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ↔ ∀ 𝑘 ∈ ( Base ‘ 𝑍 ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) |
34 |
|
ralunb |
⊢ ( ∀ 𝑘 ∈ ( 𝑈 ∪ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ↔ ( ∀ 𝑘 ∈ 𝑈 ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ∧ ∀ 𝑘 ∈ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |
35 |
33 34
|
bitr3i |
⊢ ( ∀ 𝑘 ∈ ( Base ‘ 𝑍 ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ↔ ( ∀ 𝑘 ∈ 𝑈 ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ∧ ∀ 𝑘 ∈ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |
36 |
29 35
|
bitrdi |
⊢ ( 𝜑 → ( 𝑋 = 𝑌 ↔ ( ∀ 𝑘 ∈ 𝑈 ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ∧ ∀ 𝑘 ∈ ( ( Base ‘ 𝑍 ) ∖ 𝑈 ) ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) ) |
37 |
23 36
|
mpbiran2d |
⊢ ( 𝜑 → ( 𝑋 = 𝑌 ↔ ∀ 𝑘 ∈ 𝑈 ( 𝑋 ‘ 𝑘 ) = ( 𝑌 ‘ 𝑘 ) ) ) |