Step | Hyp | Ref
| Expression |
1 | | usgreghash2spot.m |
. . . . . . . . 9
⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd
‘(1st ‘𝑡)) = 𝑎)}) |
2 | 1 | usg2spot2nb 26592 |
. . . . . . . 8
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑣 ∈ 𝑉) → (𝑀‘𝑣) = ∪ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
3 | 2 | 3expa 1257 |
. . . . . . 7
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (𝑀‘𝑣) = ∪ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
4 | 3 | fveq2d 6107 |
. . . . . 6
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (#‘(𝑀‘𝑣)) = (#‘∪ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉})) |
5 | | nbfiusgrafi 25978 |
. . . . . . . 8
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑣 ∈ 𝑉) → (〈𝑉, 𝐸〉 Neighbors 𝑣) ∈ Fin) |
6 | 5 | 3expa 1257 |
. . . . . . 7
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (〈𝑉, 𝐸〉 Neighbors 𝑣) ∈ Fin) |
7 | | diffi 8077 |
. . . . . . . . . . 11
⊢
((〈𝑉, 𝐸〉 Neighbors 𝑣) ∈ Fin →
((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ∈ Fin) |
8 | 5, 7 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑣 ∈ 𝑉) → ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ∈ Fin) |
9 | 8 | 3expa 1257 |
. . . . . . . . 9
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ∈ Fin) |
10 | 9 | adantr 480 |
. . . . . . . 8
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ∈ Fin) |
11 | | snfi 7923 |
. . . . . . . . . 10
⊢
{〈𝑐, 𝑣, 𝑑〉} ∈ Fin |
12 | 11 | a1i 11 |
. . . . . . . . 9
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) ∧ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) → {〈𝑐, 𝑣, 𝑑〉} ∈ Fin) |
13 | 12 | ralrimiva 2949 |
. . . . . . . 8
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → ∀𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} ∈ Fin) |
14 | | iunfi 8137 |
. . . . . . . 8
⊢
((((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ∈ Fin ∧ ∀𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} ∈ Fin) → ∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} ∈ Fin) |
15 | 10, 13, 14 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → ∪
𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} ∈ Fin) |
16 | | nbgrassvt 25962 |
. . . . . . . . . . . 12
⊢ (𝑉 USGrph 𝐸 → (〈𝑉, 𝐸〉 Neighbors 𝑣) ⊆ 𝑉) |
17 | 16 | ad3antrrr 762 |
. . . . . . . . . . 11
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → (〈𝑉, 𝐸〉 Neighbors 𝑣) ⊆ 𝑉) |
18 | 17 | ssdifd 3708 |
. . . . . . . . . 10
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐})) |
19 | | iunss1 4468 |
. . . . . . . . . 10
⊢
(((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}) → ∪
𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
20 | 18, 19 | syl 17 |
. . . . . . . . 9
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → ∪
𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
21 | 20 | ralrimiva 2949 |
. . . . . . . 8
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → ∀𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
22 | | simpr 476 |
. . . . . . . . 9
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) |
23 | | otiunsndisj 4905 |
. . . . . . . . 9
⊢ (𝑣 ∈ 𝑉 → Disj 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
24 | 22, 23 | syl 17 |
. . . . . . . 8
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → Disj 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
25 | | disjss2 4556 |
. . . . . . . 8
⊢
(∀𝑐 ∈
(〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} → (Disj 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉} → Disj 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉})) |
26 | 21, 24, 25 | sylc 63 |
. . . . . . 7
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → Disj 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
27 | 6, 15, 26 | hashiun 14395 |
. . . . . 6
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (#‘∪ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) = Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)(#‘∪
𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉})) |
28 | 4, 27 | eqtrd 2644 |
. . . . 5
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (#‘(𝑀‘𝑣)) = Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)(#‘∪
𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉})) |
29 | 28 | adantr 480 |
. . . 4
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘(𝑀‘𝑣)) = Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)(#‘∪
𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉})) |
30 | 9 | ad2antrr 758 |
. . . . . . 7
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ∈ Fin) |
31 | 11 | a1i 11 |
. . . . . . 7
⊢
((((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) ∧ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) → {〈𝑐, 𝑣, 𝑑〉} ∈ Fin) |
32 | | nbgraisvtx 25960 |
. . . . . . . . . 10
⊢ (𝑉 USGrph 𝐸 → (𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣) → 𝑐 ∈ 𝑉)) |
33 | 32 | ad3antrrr 762 |
. . . . . . . . 9
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣) → 𝑐 ∈ 𝑉)) |
34 | 33 | imp 444 |
. . . . . . . 8
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → 𝑐 ∈ 𝑉) |
35 | 22 | ad2antrr 758 |
. . . . . . . 8
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → 𝑣 ∈ 𝑉) |
36 | | otsndisj 4904 |
. . . . . . . 8
⊢ ((𝑐 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉) → Disj 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
37 | 34, 35, 36 | syl2anc 691 |
. . . . . . 7
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → Disj 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) |
38 | 30, 31, 37 | hashiun 14395 |
. . . . . 6
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → (#‘∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) = Σ𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})(#‘{〈𝑐, 𝑣, 𝑑〉})) |
39 | | otex 4860 |
. . . . . . . 8
⊢
〈𝑐, 𝑣, 𝑑〉 ∈ V |
40 | | hashsng 13020 |
. . . . . . . 8
⊢
(〈𝑐, 𝑣, 𝑑〉 ∈ V → (#‘{〈𝑐, 𝑣, 𝑑〉}) = 1) |
41 | 39, 40 | mp1i 13 |
. . . . . . 7
⊢
((((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) ∧ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) → (#‘{〈𝑐, 𝑣, 𝑑〉}) = 1) |
42 | 41 | sumeq2dv 14281 |
. . . . . 6
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → Σ𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})(#‘{〈𝑐, 𝑣, 𝑑〉}) = Σ𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})1) |
43 | | ax-1cn 9873 |
. . . . . . 7
⊢ 1 ∈
ℂ |
44 | | fsumconst 14364 |
. . . . . . 7
⊢
((((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}) ∈ Fin ∧ 1 ∈ ℂ) →
Σ𝑑 ∈
((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})1 = ((#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) · 1)) |
45 | 30, 43, 44 | sylancl 693 |
. . . . . 6
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → Σ𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})1 = ((#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) · 1)) |
46 | 38, 42, 45 | 3eqtrd 2648 |
. . . . 5
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → (#‘∪ 𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) = ((#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) · 1)) |
47 | 46 | sumeq2dv 14281 |
. . . 4
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)(#‘∪
𝑑 ∈ ((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐}){〈𝑐, 𝑣, 𝑑〉}) = Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)((#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) · 1)) |
48 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (〈𝑉, 𝐸〉 Neighbors 𝑣) ∈ Fin) |
49 | | hashdifsn 13063 |
. . . . . . . . 9
⊢
(((〈𝑉, 𝐸〉 Neighbors 𝑣) ∈ Fin ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → (#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) = ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) |
50 | 48, 49 | sylan 487 |
. . . . . . . 8
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → (#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) = ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) |
51 | 50 | oveq1d 6564 |
. . . . . . 7
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → ((#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) · 1) = (((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) · 1)) |
52 | | hashcl 13009 |
. . . . . . . . . . 11
⊢
((〈𝑉, 𝐸〉 Neighbors 𝑣) ∈ Fin →
(#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) ∈
ℕ0) |
53 | 6, 52 | syl 17 |
. . . . . . . . . 10
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) ∈
ℕ0) |
54 | 53 | nn0red 11229 |
. . . . . . . . 9
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) ∈ ℝ) |
55 | | peano2rem 10227 |
. . . . . . . . 9
⊢
((#‘(〈𝑉,
𝐸〉 Neighbors 𝑣)) ∈ ℝ →
((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) ∈
ℝ) |
56 | | ax-1rid 9885 |
. . . . . . . . 9
⊢
(((#‘(〈𝑉,
𝐸〉 Neighbors 𝑣)) − 1) ∈ ℝ
→ (((#‘(〈𝑉,
𝐸〉 Neighbors 𝑣)) − 1) · 1) =
((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) |
57 | 54, 55, 56 | 3syl 18 |
. . . . . . . 8
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) · 1) =
((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) |
58 | 57 | ad2antrr 758 |
. . . . . . 7
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → (((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) · 1) =
((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) |
59 | 51, 58 | eqtrd 2644 |
. . . . . 6
⊢
(((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)) → ((#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) · 1) = ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) |
60 | 59 | sumeq2dv 14281 |
. . . . 5
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)((#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) · 1) = Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) |
61 | 53 | nn0cnd 11230 |
. . . . . . . 8
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) ∈ ℂ) |
62 | 43 | a1i 11 |
. . . . . . . 8
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → 1 ∈ ℂ) |
63 | 61, 62 | subcld 10271 |
. . . . . . 7
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) ∈
ℂ) |
64 | 63 | adantr 480 |
. . . . . 6
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) ∈
ℂ) |
65 | | fsumconst 14364 |
. . . . . 6
⊢
(((〈𝑉, 𝐸〉 Neighbors 𝑣) ∈ Fin ∧
((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) ∈ ℂ)
→ Σ𝑐 ∈
(〈𝑉, 𝐸〉 Neighbors 𝑣)((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) = ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) · ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1))) |
66 | 48, 64, 65 | syl2anc 691 |
. . . . 5
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) = ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) · ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1))) |
67 | | hashnbgravdg 26440 |
. . . . . . . 8
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑣 ∈ 𝑉) → (#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) = ((𝑉 VDeg 𝐸)‘𝑣)) |
68 | | eqeq1 2614 |
. . . . . . . . . 10
⊢ (((𝑉 VDeg 𝐸)‘𝑣) = (#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ↔ (#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) = 𝐾)) |
69 | 68 | eqcoms 2618 |
. . . . . . . . 9
⊢
((#‘(〈𝑉,
𝐸〉 Neighbors 𝑣)) = ((𝑉 VDeg 𝐸)‘𝑣) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ↔ (#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) = 𝐾)) |
70 | | id 22 |
. . . . . . . . . 10
⊢
((#‘(〈𝑉,
𝐸〉 Neighbors 𝑣)) = 𝐾 → (#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) = 𝐾) |
71 | | oveq1 6556 |
. . . . . . . . . 10
⊢
((#‘(〈𝑉,
𝐸〉 Neighbors 𝑣)) = 𝐾 → ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1) = (𝐾 − 1)) |
72 | 70, 71 | oveq12d 6567 |
. . . . . . . . 9
⊢
((#‘(〈𝑉,
𝐸〉 Neighbors 𝑣)) = 𝐾 → ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) · ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1))) |
73 | 69, 72 | syl6bi 242 |
. . . . . . . 8
⊢
((#‘(〈𝑉,
𝐸〉 Neighbors 𝑣)) = ((𝑉 VDeg 𝐸)‘𝑣) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) · ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))) |
74 | 67, 73 | syl 17 |
. . . . . . 7
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑣 ∈ 𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) · ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))) |
75 | 74 | adantlr 747 |
. . . . . 6
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) · ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))) |
76 | 75 | imp 444 |
. . . . 5
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) · ((#‘(〈𝑉, 𝐸〉 Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1))) |
77 | 60, 66, 76 | 3eqtrd 2648 |
. . . 4
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑐 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑣)((#‘((〈𝑉, 𝐸〉 Neighbors 𝑣) ∖ {𝑐})) · 1) = (𝐾 · (𝐾 − 1))) |
78 | 29, 47, 77 | 3eqtrd 2648 |
. . 3
⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1))) |
79 | 78 | ex 449 |
. 2
⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ 𝑣 ∈ 𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1)))) |
80 | 79 | ralrimiva 2949 |
1
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → ∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1)))) |