Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlkn.w |
⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 ) |
2 |
|
erclwwlkn.r |
⊢ ∼ = { 〈 𝑡 , 𝑢 〉 ∣ ( 𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) } |
3 |
|
eleq1 |
⊢ ( 𝑡 = 𝑇 → ( 𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊 ) ) |
4 |
3
|
adantr |
⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( 𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊 ) ) |
5 |
|
eleq1 |
⊢ ( 𝑢 = 𝑈 → ( 𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( 𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊 ) ) |
7 |
|
simpl |
⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → 𝑡 = 𝑇 ) |
8 |
|
oveq1 |
⊢ ( 𝑢 = 𝑈 → ( 𝑢 cyclShift 𝑛 ) = ( 𝑈 cyclShift 𝑛 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( 𝑢 cyclShift 𝑛 ) = ( 𝑈 cyclShift 𝑛 ) ) |
10 |
7 9
|
eqeq12d |
⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( 𝑡 = ( 𝑢 cyclShift 𝑛 ) ↔ 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) ) |
11 |
10
|
rexbidv |
⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ↔ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) ) |
12 |
4 6 11
|
3anbi123d |
⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( ( 𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) ↔ ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) ) ) |
13 |
12 2
|
brabga |
⊢ ( ( 𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌 ) → ( 𝑇 ∼ 𝑈 ↔ ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) ) ) |