Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlk.r |
⊢ ∼ = { 〈 𝑢 , 𝑤 〉 ∣ ( 𝑢 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 𝑢 = ( 𝑤 cyclShift 𝑛 ) ) } |
2 |
|
eleq1 |
⊢ ( 𝑢 = 𝑈 → ( 𝑢 ∈ ( ClWWalks ‘ 𝐺 ) ↔ 𝑈 ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
3 |
2
|
adantr |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑤 = 𝑊 ) → ( 𝑢 ∈ ( ClWWalks ‘ 𝐺 ) ↔ 𝑈 ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
4 |
|
eleq1 |
⊢ ( 𝑤 = 𝑊 → ( 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ↔ 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑤 = 𝑊 ) → ( 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ↔ 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ) ) |
6 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ 𝑊 ) ) |
7 |
6
|
oveq2d |
⊢ ( 𝑤 = 𝑊 → ( 0 ... ( ♯ ‘ 𝑤 ) ) = ( 0 ... ( ♯ ‘ 𝑊 ) ) ) |
8 |
7
|
adantl |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑤 = 𝑊 ) → ( 0 ... ( ♯ ‘ 𝑤 ) ) = ( 0 ... ( ♯ ‘ 𝑊 ) ) ) |
9 |
|
simpl |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑤 = 𝑊 ) → 𝑢 = 𝑈 ) |
10 |
|
oveq1 |
⊢ ( 𝑤 = 𝑊 → ( 𝑤 cyclShift 𝑛 ) = ( 𝑊 cyclShift 𝑛 ) ) |
11 |
10
|
adantl |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑤 = 𝑊 ) → ( 𝑤 cyclShift 𝑛 ) = ( 𝑊 cyclShift 𝑛 ) ) |
12 |
9 11
|
eqeq12d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑤 = 𝑊 ) → ( 𝑢 = ( 𝑤 cyclShift 𝑛 ) ↔ 𝑈 = ( 𝑊 cyclShift 𝑛 ) ) ) |
13 |
8 12
|
rexeqbidv |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑤 = 𝑊 ) → ( ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 𝑢 = ( 𝑤 cyclShift 𝑛 ) ↔ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) 𝑈 = ( 𝑊 cyclShift 𝑛 ) ) ) |
14 |
3 5 13
|
3anbi123d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑤 = 𝑊 ) → ( ( 𝑢 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 𝑢 = ( 𝑤 cyclShift 𝑛 ) ) ↔ ( 𝑈 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) 𝑈 = ( 𝑊 cyclShift 𝑛 ) ) ) ) |
15 |
14 1
|
brabga |
⊢ ( ( 𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → ( 𝑈 ∼ 𝑊 ↔ ( 𝑈 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) 𝑈 = ( 𝑊 cyclShift 𝑛 ) ) ) ) |