Step |
Hyp |
Ref |
Expression |
1 |
|
wlkp1.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
wlkp1.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
wlkp1.f |
⊢ ( 𝜑 → Fun 𝐼 ) |
4 |
|
wlkp1.a |
⊢ ( 𝜑 → 𝐼 ∈ Fin ) |
5 |
|
wlkp1.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
6 |
|
wlkp1.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
7 |
|
wlkp1.d |
⊢ ( 𝜑 → ¬ 𝐵 ∈ dom 𝐼 ) |
8 |
|
wlkp1.w |
⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
9 |
|
wlkp1.n |
⊢ 𝑁 = ( ♯ ‘ 𝐹 ) |
10 |
|
wlkp1.e |
⊢ ( 𝜑 → 𝐸 ∈ ( Edg ‘ 𝐺 ) ) |
11 |
|
wlkp1.x |
⊢ ( 𝜑 → { ( 𝑃 ‘ 𝑁 ) , 𝐶 } ⊆ 𝐸 ) |
12 |
|
wlkp1.u |
⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐼 ∪ { 〈 𝐵 , 𝐸 〉 } ) ) |
13 |
|
wlkp1.h |
⊢ 𝐻 = ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) |
14 |
13
|
fveq2i |
⊢ ( ♯ ‘ 𝐻 ) = ( ♯ ‘ ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) ) |
15 |
14
|
a1i |
⊢ ( 𝜑 → ( ♯ ‘ 𝐻 ) = ( ♯ ‘ ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) ) ) |
16 |
|
opex |
⊢ 〈 𝑁 , 𝐵 〉 ∈ V |
17 |
2
|
wlkf |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom 𝐼 ) |
18 |
|
wrdfin |
⊢ ( 𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ Fin ) |
19 |
8 17 18
|
3syl |
⊢ ( 𝜑 → 𝐹 ∈ Fin ) |
20 |
|
fzonel |
⊢ ¬ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) |
21 |
20
|
a1i |
⊢ ( 𝜑 → ¬ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
22 |
|
eleq1 |
⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
23 |
22
|
notbid |
⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( ¬ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↔ ¬ ( ♯ ‘ 𝐹 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
24 |
21 23
|
syl5ibr |
⊢ ( 𝑁 = ( ♯ ‘ 𝐹 ) → ( 𝜑 → ¬ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
25 |
9 24
|
ax-mp |
⊢ ( 𝜑 → ¬ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
26 |
|
wrdfn |
⊢ ( 𝐹 ∈ Word dom 𝐼 → 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
27 |
8 17 26
|
3syl |
⊢ ( 𝜑 → 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
28 |
|
fnop |
⊢ ( ( 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ∧ 〈 𝑁 , 𝐵 〉 ∈ 𝐹 ) → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
29 |
28
|
ex |
⊢ ( 𝐹 Fn ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 〈 𝑁 , 𝐵 〉 ∈ 𝐹 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
30 |
27 29
|
syl |
⊢ ( 𝜑 → ( 〈 𝑁 , 𝐵 〉 ∈ 𝐹 → 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) |
31 |
25 30
|
mtod |
⊢ ( 𝜑 → ¬ 〈 𝑁 , 𝐵 〉 ∈ 𝐹 ) |
32 |
19 31
|
jca |
⊢ ( 𝜑 → ( 𝐹 ∈ Fin ∧ ¬ 〈 𝑁 , 𝐵 〉 ∈ 𝐹 ) ) |
33 |
|
hashunsng |
⊢ ( 〈 𝑁 , 𝐵 〉 ∈ V → ( ( 𝐹 ∈ Fin ∧ ¬ 〈 𝑁 , 𝐵 〉 ∈ 𝐹 ) → ( ♯ ‘ ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) ) |
34 |
16 32 33
|
mpsyl |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝐹 ∪ { 〈 𝑁 , 𝐵 〉 } ) ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) |
35 |
9
|
eqcomi |
⊢ ( ♯ ‘ 𝐹 ) = 𝑁 |
36 |
35
|
a1i |
⊢ ( 𝜑 → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
37 |
36
|
oveq1d |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐹 ) + 1 ) = ( 𝑁 + 1 ) ) |
38 |
15 34 37
|
3eqtrd |
⊢ ( 𝜑 → ( ♯ ‘ 𝐻 ) = ( 𝑁 + 1 ) ) |