Step |
Hyp |
Ref |
Expression |
1 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
2 |
|
wwlksn |
⊢ ( 0 ∈ ℕ0 → ( 0 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 0 + 1 ) } ) |
3 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
5 |
3 4
|
iswwlks |
⊢ ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ↔ ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
6 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
7 |
6
|
eqeq2i |
⊢ ( ( ♯ ‘ 𝑤 ) = ( 0 + 1 ) ↔ ( ♯ ‘ 𝑤 ) = 1 ) |
8 |
5 7
|
anbi12i |
⊢ ( ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 0 + 1 ) ) ↔ ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ) |
9 |
|
simp2 |
⊢ ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) → 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ) |
10 |
|
vex |
⊢ 𝑤 ∈ V |
11 |
|
0lt1 |
⊢ 0 < 1 |
12 |
|
breq2 |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ( 0 < ( ♯ ‘ 𝑤 ) ↔ 0 < 1 ) ) |
13 |
11 12
|
mpbiri |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → 0 < ( ♯ ‘ 𝑤 ) ) |
14 |
|
hashgt0n0 |
⊢ ( ( 𝑤 ∈ V ∧ 0 < ( ♯ ‘ 𝑤 ) ) → 𝑤 ≠ ∅ ) |
15 |
10 13 14
|
sylancr |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → 𝑤 ≠ ∅ ) |
16 |
15
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑤 ) = 1 ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ) → 𝑤 ≠ ∅ ) |
17 |
|
simpr |
⊢ ( ( ( ♯ ‘ 𝑤 ) = 1 ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ) → 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ) |
18 |
|
ral0 |
⊢ ∀ 𝑖 ∈ ∅ { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) |
19 |
|
oveq1 |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ( ( ♯ ‘ 𝑤 ) − 1 ) = ( 1 − 1 ) ) |
20 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
21 |
19 20
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ( ( ♯ ‘ 𝑤 ) − 1 ) = 0 ) |
22 |
21
|
oveq2d |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) = ( 0 ..^ 0 ) ) |
23 |
|
fzo0 |
⊢ ( 0 ..^ 0 ) = ∅ |
24 |
22 23
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) = ∅ ) |
25 |
24
|
raleqdv |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ ∀ 𝑖 ∈ ∅ { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
26 |
18 25
|
mpbiri |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) |
27 |
26
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑤 ) = 1 ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) |
28 |
16 17 27
|
3jca |
⊢ ( ( ( ♯ ‘ 𝑤 ) = 1 ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
29 |
28
|
ex |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) → ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
30 |
9 29
|
impbid2 |
⊢ ( ( ♯ ‘ 𝑤 ) = 1 → ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ↔ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ) ) |
31 |
30
|
pm5.32ri |
⊢ ( ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ↔ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ) |
32 |
8 31
|
bitri |
⊢ ( ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 0 + 1 ) ) ↔ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ) |
33 |
32
|
a1i |
⊢ ( 0 ∈ ℕ0 → ( ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 0 + 1 ) ) ↔ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ) ) |
34 |
33
|
rabbidva2 |
⊢ ( 0 ∈ ℕ0 → { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 0 + 1 ) } = { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 1 } ) |
35 |
2 34
|
eqtrd |
⊢ ( 0 ∈ ℕ0 → ( 0 WWalksN 𝐺 ) = { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 1 } ) |
36 |
1 35
|
ax-mp |
⊢ ( 0 WWalksN 𝐺 ) = { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 1 } |