Step |
Hyp |
Ref |
Expression |
1 |
|
2nn |
⊢ 2 ∈ ℕ |
2 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
4 |
2 3
|
isclwwlknx |
⊢ ( 2 ∈ ℕ → ( 𝑊 ∈ ( 2 ClWWalksN 𝐺 ) ↔ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑊 ) = 2 ) ) ) |
5 |
1 4
|
ax-mp |
⊢ ( 𝑊 ∈ ( 2 ClWWalksN 𝐺 ) ↔ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑊 ) = 2 ) ) |
6 |
|
3anass |
⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ↔ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
7 |
|
oveq1 |
⊢ ( ( ♯ ‘ 𝑊 ) = 2 → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( 2 − 1 ) ) |
8 |
|
2m1e1 |
⊢ ( 2 − 1 ) = 1 |
9 |
7 8
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝑊 ) = 2 → ( ( ♯ ‘ 𝑊 ) − 1 ) = 1 ) |
10 |
9
|
oveq2d |
⊢ ( ( ♯ ‘ 𝑊 ) = 2 → ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 0 ..^ 1 ) ) |
11 |
|
fzo01 |
⊢ ( 0 ..^ 1 ) = { 0 } |
12 |
10 11
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝑊 ) = 2 → ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = { 0 } ) |
13 |
12
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = { 0 } ) |
14 |
13
|
raleqdv |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ ∀ 𝑖 ∈ { 0 } { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
15 |
|
c0ex |
⊢ 0 ∈ V |
16 |
|
fveq2 |
⊢ ( 𝑖 = 0 → ( 𝑊 ‘ 𝑖 ) = ( 𝑊 ‘ 0 ) ) |
17 |
|
fv0p1e1 |
⊢ ( 𝑖 = 0 → ( 𝑊 ‘ ( 𝑖 + 1 ) ) = ( 𝑊 ‘ 1 ) ) |
18 |
16 17
|
preq12d |
⊢ ( 𝑖 = 0 → { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } = { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ) |
19 |
18
|
eleq1d |
⊢ ( 𝑖 = 0 → ( { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
20 |
15 19
|
ralsn |
⊢ ( ∀ 𝑖 ∈ { 0 } { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) |
21 |
14 20
|
bitrdi |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
22 |
|
prcom |
⊢ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } = { ( 𝑊 ‘ 0 ) , ( lastS ‘ 𝑊 ) } |
23 |
|
lsw |
⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
24 |
9
|
fveq2d |
⊢ ( ( ♯ ‘ 𝑊 ) = 2 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ 1 ) ) |
25 |
23 24
|
sylan9eqr |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ 1 ) ) |
26 |
25
|
preq2d |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → { ( 𝑊 ‘ 0 ) , ( lastS ‘ 𝑊 ) } = { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ) |
27 |
22 26
|
syl5eq |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } = { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ) |
28 |
27
|
eleq1d |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ↔ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
29 |
21 28
|
anbi12d |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ↔ ( { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
30 |
|
anidm |
⊢ ( ( { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ↔ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) |
31 |
29 30
|
bitrdi |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ↔ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
32 |
31
|
pm5.32da |
⊢ ( ( ♯ ‘ 𝑊 ) = 2 → ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ↔ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
33 |
6 32
|
syl5bb |
⊢ ( ( ♯ ‘ 𝑊 ) = 2 → ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ↔ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
34 |
33
|
pm5.32ri |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑊 ) = 2 ) ↔ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑊 ) = 2 ) ) |
35 |
|
3anass |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ↔ ( ( ♯ ‘ 𝑊 ) = 2 ∧ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
36 |
|
ancom |
⊢ ( ( ( ♯ ‘ 𝑊 ) = 2 ∧ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ↔ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑊 ) = 2 ) ) |
37 |
35 36
|
bitr2i |
⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑊 ) = 2 ) ↔ ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
38 |
5 34 37
|
3bitri |
⊢ ( 𝑊 ∈ ( 2 ClWWalksN 𝐺 ) ↔ ( ( ♯ ‘ 𝑊 ) = 2 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 1 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |