Step | Hyp | Ref
| Expression |
1 | | frgrhash2wsp.v |
. . . . . . . 8
⊢ 𝑉 = (Vtx‘𝐺) |
2 | | fusgreg2wsp.m |
. . . . . . . 8
⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) |
3 | 1, 2 | fusgr2wsp2nb 41498 |
. . . . . . 7
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (𝑀‘𝑣) = ∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
4 | 3 | fveq2d 6107 |
. . . . . 6
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (#‘(𝑀‘𝑣)) = (#‘∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉})) |
5 | 1 | eleq2i 2680 |
. . . . . . . 8
⊢ (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtx‘𝐺)) |
6 | | nbfiusgrfi 40603 |
. . . . . . . 8
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑣) ∈ Fin) |
7 | 5, 6 | sylan2b 491 |
. . . . . . 7
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (𝐺 NeighbVtx 𝑣) ∈ Fin) |
8 | 7 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ∈ Fin) |
9 | | diffi 8077 |
. . . . . . . . 9
⊢ ((𝐺 NeighbVtx 𝑣) ∈ Fin → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin) |
10 | 8, 9 | syl 17 |
. . . . . . . 8
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin) |
11 | | snfi 7923 |
. . . . . . . . . 10
⊢
{〈“𝑐𝑣𝑑”〉} ∈ Fin |
12 | 11 | a1i 11 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → {〈“𝑐𝑣𝑑”〉} ∈ Fin) |
13 | 12 | ralrimiva 2949 |
. . . . . . . 8
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ∀𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} ∈ Fin) |
14 | | iunfi 8137 |
. . . . . . . 8
⊢ ((((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin ∧ ∀𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} ∈ Fin) → ∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} ∈ Fin) |
15 | 10, 13, 14 | syl2anc 691 |
. . . . . . 7
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ∪
𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} ∈ Fin) |
16 | 1 | nbgrssvtx 40582 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ FinUSGraph → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉) |
17 | 16 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉) |
18 | 17 | ssdifd 3708 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐})) |
19 | | iunss1 4468 |
. . . . . . . . . 10
⊢ (((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}) → ∪
𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
20 | 18, 19 | syl 17 |
. . . . . . . . 9
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ∪
𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
21 | 20 | ralrimiva 2949 |
. . . . . . . 8
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → ∀𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
22 | | simpr 476 |
. . . . . . . . 9
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) |
23 | | s3iunsndisj 13555 |
. . . . . . . . 9
⊢ (𝑣 ∈ 𝑉 → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
24 | 22, 23 | syl 17 |
. . . . . . . 8
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
25 | | disjss2 4556 |
. . . . . . . 8
⊢
(∀𝑐 ∈
(𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} → (Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){〈“𝑐𝑣𝑑”〉} → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉})) |
26 | 21, 24, 25 | sylc 63 |
. . . . . . 7
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
27 | 7, 15, 26 | hashiun 14395 |
. . . . . 6
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (#‘∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)(#‘∪
𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉})) |
28 | 4, 27 | eqtrd 2644 |
. . . . 5
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (#‘(𝑀‘𝑣)) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)(#‘∪
𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉})) |
29 | 28 | adantr 480 |
. . . 4
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (#‘(𝑀‘𝑣)) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)(#‘∪
𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉})) |
30 | 7, 9 | syl 17 |
. . . . . . . 8
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin) |
31 | 30 | ad2antrr 758 |
. . . . . . 7
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin) |
32 | 11 | a1i 11 |
. . . . . . 7
⊢
(((((𝐺 ∈
FinUSGraph ∧ 𝑣 ∈
𝑉) ∧
((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → {〈“𝑐𝑣𝑑”〉} ∈ Fin) |
33 | 22 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝑣 ∈ 𝑉) |
34 | 33 | anim1i 590 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑣 ∈ 𝑉 ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣))) |
35 | 34 | ancomd 466 |
. . . . . . . 8
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉)) |
36 | | s3sndisj 13554 |
. . . . . . . 8
⊢ ((𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
37 | 35, 36 | syl 17 |
. . . . . . 7
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) |
38 | 31, 32, 37 | hashiun 14395 |
. . . . . 6
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (#‘∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) = Σ𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})(#‘{〈“𝑐𝑣𝑑”〉})) |
39 | | s3cli 13476 |
. . . . . . . 8
⊢
〈“𝑐𝑣𝑑”〉 ∈ Word V |
40 | | hashsng 13020 |
. . . . . . . 8
⊢
(〈“𝑐𝑣𝑑”〉 ∈ Word V →
(#‘{〈“𝑐𝑣𝑑”〉}) = 1) |
41 | 39, 40 | mp1i 13 |
. . . . . . 7
⊢
(((((𝐺 ∈
FinUSGraph ∧ 𝑣 ∈
𝑉) ∧
((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → (#‘{〈“𝑐𝑣𝑑”〉}) = 1) |
42 | 41 | sumeq2dv 14281 |
. . . . . 6
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Σ𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})(#‘{〈“𝑐𝑣𝑑”〉}) = Σ𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})1) |
43 | | ax-1cn 9873 |
. . . . . . 7
⊢ 1 ∈
ℂ |
44 | | fsumconst 14364 |
. . . . . . 7
⊢ ((((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin ∧ 1 ∈ ℂ) →
Σ𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})1 = ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1)) |
45 | 31, 43, 44 | sylancl 693 |
. . . . . 6
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Σ𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})1 = ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1)) |
46 | 38, 42, 45 | 3eqtrd 2648 |
. . . . 5
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (#‘∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) = ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1)) |
47 | 46 | sumeq2dv 14281 |
. . . 4
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)(#‘∪
𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){〈“𝑐𝑣𝑑”〉}) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1)) |
48 | 7 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 NeighbVtx 𝑣) ∈ Fin) |
49 | | hashdifsn 13063 |
. . . . . . . . 9
⊢ (((𝐺 NeighbVtx 𝑣) ∈ Fin ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) |
50 | 48, 49 | sylan 487 |
. . . . . . . 8
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) |
51 | 50 | oveq1d 6564 |
. . . . . . 7
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1) = (((#‘(𝐺 NeighbVtx 𝑣)) − 1) · 1)) |
52 | 1 | hashnbusgrnn0 40604 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) ∈
ℕ0) |
53 | 52 | nn0red 11229 |
. . . . . . . . 9
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) ∈ ℝ) |
54 | | peano2rem 10227 |
. . . . . . . . 9
⊢
((#‘(𝐺
NeighbVtx 𝑣)) ∈
ℝ → ((#‘(𝐺
NeighbVtx 𝑣)) − 1)
∈ ℝ) |
55 | | ax-1rid 9885 |
. . . . . . . . 9
⊢
(((#‘(𝐺
NeighbVtx 𝑣)) − 1)
∈ ℝ → (((#‘(𝐺 NeighbVtx 𝑣)) − 1) · 1) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) |
56 | 53, 54, 55 | 3syl 18 |
. . . . . . . 8
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (((#‘(𝐺 NeighbVtx 𝑣)) − 1) · 1) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) |
57 | 56 | ad2antrr 758 |
. . . . . . 7
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (((#‘(𝐺 NeighbVtx 𝑣)) − 1) · 1) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) |
58 | 51, 57 | eqtrd 2644 |
. . . . . 6
⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) |
59 | 58 | sumeq2dv 14281 |
. . . . 5
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘(𝐺 NeighbVtx 𝑣)) − 1)) |
60 | 52 | nn0cnd 11230 |
. . . . . . . 8
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) ∈ ℂ) |
61 | | 1cnd 9935 |
. . . . . . . 8
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → 1 ∈ ℂ) |
62 | 60, 61 | subcld 10271 |
. . . . . . 7
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → ((#‘(𝐺 NeighbVtx 𝑣)) − 1) ∈
ℂ) |
63 | 62 | adantr 480 |
. . . . . 6
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((#‘(𝐺 NeighbVtx 𝑣)) − 1) ∈
ℂ) |
64 | | fsumconst 14364 |
. . . . . 6
⊢ (((𝐺 NeighbVtx 𝑣) ∈ Fin ∧ ((#‘(𝐺 NeighbVtx 𝑣)) − 1) ∈ ℂ) →
Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘(𝐺 NeighbVtx 𝑣)) − 1) = ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1))) |
65 | 48, 63, 64 | syl2anc 691 |
. . . . 5
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘(𝐺 NeighbVtx 𝑣)) − 1) = ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1))) |
66 | | fusgrusgr 40541 |
. . . . . . . 8
⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph
) |
67 | 1 | hashnbusgrvd 40744 |
. . . . . . . 8
⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)) |
68 | 66, 67 | sylan 487 |
. . . . . . 7
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣)) |
69 | | eqeq1 2614 |
. . . . . . . . 9
⊢
(((VtxDeg‘𝐺)‘𝑣) = (#‘(𝐺 NeighbVtx 𝑣)) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ (#‘(𝐺 NeighbVtx 𝑣)) = 𝐾)) |
70 | 69 | eqcoms 2618 |
. . . . . . . 8
⊢
((#‘(𝐺
NeighbVtx 𝑣)) =
((VtxDeg‘𝐺)‘𝑣) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ (#‘(𝐺 NeighbVtx 𝑣)) = 𝐾)) |
71 | | id 22 |
. . . . . . . . 9
⊢
((#‘(𝐺
NeighbVtx 𝑣)) = 𝐾 → (#‘(𝐺 NeighbVtx 𝑣)) = 𝐾) |
72 | | oveq1 6556 |
. . . . . . . . 9
⊢
((#‘(𝐺
NeighbVtx 𝑣)) = 𝐾 → ((#‘(𝐺 NeighbVtx 𝑣)) − 1) = (𝐾 − 1)) |
73 | 71, 72 | oveq12d 6567 |
. . . . . . . 8
⊢
((#‘(𝐺
NeighbVtx 𝑣)) = 𝐾 → ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1))) |
74 | 70, 73 | syl6bi 242 |
. . . . . . 7
⊢
((#‘(𝐺
NeighbVtx 𝑣)) =
((VtxDeg‘𝐺)‘𝑣) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))) |
75 | 68, 74 | syl 17 |
. . . . . 6
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))) |
76 | 75 | imp 444 |
. . . . 5
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1))) |
77 | 59, 65, 76 | 3eqtrd 2648 |
. . . 4
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1) = (𝐾 · (𝐾 − 1))) |
78 | 29, 47, 77 | 3eqtrd 2648 |
. . 3
⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (#‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1))) |
79 | 78 | ex 449 |
. 2
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1)))) |
80 | 79 | ralrimiva 2949 |
1
⊢ (𝐺 ∈ FinUSGraph →
∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1)))) |