Step | Hyp | Ref
| Expression |
1 | | rusgrusgr 27506 |
. . . . . 6
⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph) |
2 | | usgruspgr 27123 |
. . . . . 6
⊢ (𝐺 ∈ USGraph → 𝐺 ∈
USPGraph) |
3 | 1, 2 | syl 17 |
. . . . 5
⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USPGraph) |
4 | 3 | ad2antlr 727 |
. . . 4
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝐺 ∈
USPGraph) |
5 | | simpl 486 |
. . . . 5
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ 𝑋 ∈ 𝑉) |
6 | 5 | adantl 485 |
. . . 4
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝑋 ∈ 𝑉) |
7 | | uzuzle23 12371 |
. . . . 5
⊢ (𝑁 ∈
(ℤ≥‘3) → 𝑁 ∈
(ℤ≥‘2)) |
8 | 7 | ad2antll 729 |
. . . 4
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝑁 ∈
(ℤ≥‘2)) |
9 | | numclwlk1.v |
. . . . 5
⊢ 𝑉 = (Vtx‘𝐺) |
10 | | numclwlk1.c |
. . . . 5
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st
‘𝑤)) = 𝑁 ∧ ((2nd
‘𝑤)‘0) = 𝑋 ∧ ((2nd
‘𝑤)‘(𝑁 − 2)) = 𝑋)} |
11 | | eqid 2738 |
. . . . 5
⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} |
12 | 9, 10, 11 | dlwwlknondlwlknonen 28303 |
. . . 4
⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))
→ 𝐶 ≈ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
13 | 4, 6, 8, 12 | syl3anc 1372 |
. . 3
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝐶 ≈ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
14 | 1 | anim2i 620 |
. . . . . . . . 9
⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph)) |
15 | 14 | ancomd 465 |
. . . . . . . 8
⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
16 | 9 | isfusgr 27260 |
. . . . . . . 8
⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
17 | 15, 16 | sylibr 237 |
. . . . . . 7
⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph) |
18 | | eluzge3nn 12372 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘3) → 𝑁 ∈ ℕ) |
19 | 18 | nnnn0d 12036 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘3) → 𝑁 ∈
ℕ0) |
20 | 19 | adantl 485 |
. . . . . . 7
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ 𝑁 ∈
ℕ0) |
21 | | wlksnfi 27845 |
. . . . . . 7
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0)
→ {𝑤 ∈
(Walks‘𝐺) ∣
(♯‘(1st ‘𝑤)) = 𝑁} ∈ Fin) |
22 | 17, 20, 21 | syl2an 599 |
. . . . . 6
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈
(Walks‘𝐺) ∣
(♯‘(1st ‘𝑤)) = 𝑁} ∈ Fin) |
23 | | clwlkswks 27717 |
. . . . . . . 8
⊢
(ClWalks‘𝐺)
⊆ (Walks‘𝐺) |
24 | 23 | a1i 11 |
. . . . . . 7
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (ClWalks‘𝐺)
⊆ (Walks‘𝐺)) |
25 | | simp21 1207 |
. . . . . . 7
⊢ ((((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
∧ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → (♯‘(1st
‘𝑤)) = 𝑁) |
26 | 24, 25 | rabssrabd 3972 |
. . . . . 6
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈
(ClWalks‘𝐺) ∣
((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} ⊆ {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁}) |
27 | 22, 26 | ssfid 8819 |
. . . . 5
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈
(ClWalks‘𝐺) ∣
((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} ∈ Fin) |
28 | 10, 27 | eqeltrid 2837 |
. . . 4
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝐶 ∈
Fin) |
29 | 9 | clwwlknonfin 28031 |
. . . . . 6
⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ Fin) |
30 | 29 | ad2antrr 726 |
. . . . 5
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ Fin) |
31 | | ssrab2 3969 |
. . . . . 6
⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ⊆ (𝑋(ClWWalksNOn‘𝐺)𝑁) |
32 | 31 | a1i 11 |
. . . . 5
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ⊆ (𝑋(ClWWalksNOn‘𝐺)𝑁)) |
33 | 30, 32 | ssfid 8819 |
. . . 4
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ∈ Fin) |
34 | | hashen 13799 |
. . . 4
⊢ ((𝐶 ∈ Fin ∧ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ∈ Fin) → ((♯‘𝐶) = (♯‘{𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) ↔ 𝐶 ≈ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})) |
35 | 28, 33, 34 | syl2anc 587 |
. . 3
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ ((♯‘𝐶) =
(♯‘{𝑤 ∈
(𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) ↔ 𝐶 ≈ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})) |
36 | 13, 35 | mpbird 260 |
. 2
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (♯‘𝐶) =
(♯‘{𝑤 ∈
(𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})) |
37 | | eqidd 2739 |
. . . 4
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})) |
38 | | oveq12 7179 |
. . . . . 6
⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑣(ClWWalksNOn‘𝐺)𝑛) = (𝑋(ClWWalksNOn‘𝐺)𝑁)) |
39 | | fvoveq1 7193 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2))) |
40 | 39 | adantl 485 |
. . . . . . 7
⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑤‘(𝑛 − 2)) = (𝑤‘(𝑁 − 2))) |
41 | | simpl 486 |
. . . . . . 7
⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → 𝑣 = 𝑋) |
42 | 40, 41 | eqeq12d 2754 |
. . . . . 6
⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → ((𝑤‘(𝑛 − 2)) = 𝑣 ↔ (𝑤‘(𝑁 − 2)) = 𝑋)) |
43 | 38, 42 | rabeqbidv 3387 |
. . . . 5
⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣} = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
44 | 43 | adantl 485 |
. . . 4
⊢ ((((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
∧ (𝑣 = 𝑋 ∧ 𝑛 = 𝑁)) → {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣} = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
45 | | ovex 7203 |
. . . . . 6
⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ V |
46 | 45 | rabex 5200 |
. . . . 5
⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ∈ V |
47 | 46 | a1i 11 |
. . . 4
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ∈ V) |
48 | 37, 44, 6, 8, 47 | ovmpod 7317 |
. . 3
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) |
49 | 48 | fveq2d 6678 |
. 2
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁)) = (♯‘{𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})) |
50 | | eqid 2738 |
. . . 4
⊢ (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) |
51 | | eqid 2738 |
. . . 4
⊢ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) |
52 | 9, 50, 51 | numclwwlk1 28298 |
. . 3
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁)) = (𝐾 · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))))) |
53 | | numclwlk1.f |
. . . . . . 7
⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st
‘𝑤)) = (𝑁 − 2) ∧
((2nd ‘𝑤)‘0) = 𝑋)} |
54 | 5, 9 | eleqtrdi 2843 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ 𝑋 ∈
(Vtx‘𝐺)) |
55 | 54 | adantl 485 |
. . . . . . . 8
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝑋 ∈
(Vtx‘𝐺)) |
56 | | uz3m2nn 12373 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) |
57 | 56 | ad2antll 729 |
. . . . . . . 8
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (𝑁 − 2) ∈
ℕ) |
58 | | clwwlknonclwlknonen 28300 |
. . . . . . . 8
⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ (𝑁 − 2) ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣
((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd
‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) |
59 | 4, 55, 57, 58 | syl3anc 1372 |
. . . . . . 7
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈
(ClWalks‘𝐺) ∣
((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd
‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) |
60 | 53, 59 | eqbrtrid 5065 |
. . . . . 6
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝐹 ≈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) |
61 | | uznn0sub 12359 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈
(ℤ≥‘2) → (𝑁 − 2) ∈
ℕ0) |
62 | 7, 61 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘3) → (𝑁 − 2) ∈
ℕ0) |
63 | 62 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (𝑁 − 2) ∈
ℕ0) |
64 | | wlksnfi 27845 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ FinUSGraph ∧ (𝑁 − 2) ∈
ℕ0) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = (𝑁 − 2)} ∈
Fin) |
65 | 17, 63, 64 | syl2an 599 |
. . . . . . . . 9
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈
(Walks‘𝐺) ∣
(♯‘(1st ‘𝑤)) = (𝑁 − 2)} ∈ Fin) |
66 | | simp2l 1200 |
. . . . . . . . . 10
⊢ ((((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
∧ ((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd
‘𝑤)‘0) = 𝑋) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → (♯‘(1st
‘𝑤)) = (𝑁 − 2)) |
67 | 24, 66 | rabssrabd 3972 |
. . . . . . . . 9
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈
(ClWalks‘𝐺) ∣
((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd
‘𝑤)‘0) = 𝑋)} ⊆ {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = (𝑁 − 2)}) |
68 | 65, 67 | ssfid 8819 |
. . . . . . . 8
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ {𝑤 ∈
(ClWalks‘𝐺) ∣
((♯‘(1st ‘𝑤)) = (𝑁 − 2) ∧ ((2nd
‘𝑤)‘0) = 𝑋)} ∈ Fin) |
69 | 53, 68 | eqeltrid 2837 |
. . . . . . 7
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝐹 ∈
Fin) |
70 | 9 | clwwlknonfin 28031 |
. . . . . . . 8
⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin) |
71 | 70 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin) |
72 | | hashen 13799 |
. . . . . . 7
⊢ ((𝐹 ∈ Fin ∧ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ∈ Fin) →
((♯‘𝐹) =
(♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) ↔ 𝐹 ≈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) |
73 | 69, 71, 72 | syl2anc 587 |
. . . . . 6
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ ((♯‘𝐹) =
(♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) ↔ 𝐹 ≈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) |
74 | 60, 73 | mpbird 260 |
. . . . 5
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (♯‘𝐹) =
(♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) |
75 | 74 | eqcomd 2744 |
. . . 4
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) = (♯‘𝐹)) |
76 | 75 | oveq2d 7186 |
. . 3
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (𝐾 ·
(♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) = (𝐾 · (♯‘𝐹))) |
77 | 52, 76 | eqtrd 2773 |
. 2
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (♯‘(𝑋(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})𝑁)) = (𝐾 · (♯‘𝐹))) |
78 | 36, 49, 77 | 3eqtr2d 2779 |
1
⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (♯‘𝐶) =
(𝐾 ·
(♯‘𝐹))) |