Step |
Hyp |
Ref |
Expression |
1 |
|
3wlkd.p |
⊢ 𝑃 = 〈“ 𝐴 𝐵 𝐶 𝐷 ”〉 |
2 |
|
3wlkd.f |
⊢ 𝐹 = 〈“ 𝐽 𝐾 𝐿 ”〉 |
3 |
|
3wlkd.s |
⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) ) |
4 |
|
3wlkd.n |
⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) |
5 |
|
3wlkd.e |
⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) ) |
6 |
|
3wlkd.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
7 |
|
3wlkd.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
8 |
|
s4cli |
⊢ 〈“ 𝐴 𝐵 𝐶 𝐷 ”〉 ∈ Word V |
9 |
1 8
|
eqeltri |
⊢ 𝑃 ∈ Word V |
10 |
9
|
a1i |
⊢ ( 𝜑 → 𝑃 ∈ Word V ) |
11 |
|
s3cli |
⊢ 〈“ 𝐽 𝐾 𝐿 ”〉 ∈ Word V |
12 |
2 11
|
eqeltri |
⊢ 𝐹 ∈ Word V |
13 |
12
|
a1i |
⊢ ( 𝜑 → 𝐹 ∈ Word V ) |
14 |
1 2
|
3wlkdlem1 |
⊢ ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) |
15 |
14
|
a1i |
⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) |
16 |
1 2 3 4 5
|
3wlkdlem10 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
17 |
1 2 3 4
|
3wlkdlem5 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) |
18 |
6
|
1vgrex |
⊢ ( 𝐴 ∈ 𝑉 → 𝐺 ∈ V ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → 𝐺 ∈ V ) |
20 |
3 19
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ V ) |
21 |
1 2 3
|
3wlkdlem4 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 ) |
22 |
10 13 15 16 17 20 6 7 21
|
wlkd |
⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |