Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fusgreghash2wspv Structured version   Visualization version   GIF version

Theorem fusgreghash2wspv 41499
 Description: According to statement 7 in [Huneke] p. 2: "For each vertex v, there are exactly ( k 2 ) paths with length two having v in the middle, ..." in a finite k-regular graph. For simple paths of length 2 represented by length 3 strings, we have again k*(k-1) such paths. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 17-May-2021.)
Hypotheses
Ref Expression
frgrhash2wsp.v 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
Assertion
Ref Expression
fusgreghash2wspv (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
Distinct variable groups:   𝐺,𝑎   𝑉,𝑎   𝑤,𝐺,𝑎   𝑤,𝑉   𝑣,𝐺,𝑎,𝑤
Allowed substitution hints:   𝐾(𝑤,𝑣,𝑎)   𝑀(𝑤,𝑣,𝑎)   𝑉(𝑣)

Proof of Theorem fusgreghash2wspv
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrhash2wsp.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
2 fusgreg2wsp.m . . . . . . . 8 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
31, 2fusgr2wsp2nb 41498 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (𝑀𝑣) = 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
43fveq2d 6107 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (#‘(𝑀𝑣)) = (#‘ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
51eleq2i 2680 . . . . . . . 8 (𝑣𝑉𝑣 ∈ (Vtx‘𝐺))
6 nbfiusgrfi 40603 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
75, 6sylan2b 491 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
87adantr 480 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
9 diffi 8077 . . . . . . . . 9 ((𝐺 NeighbVtx 𝑣) ∈ Fin → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin)
108, 9syl 17 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin)
11 snfi 7923 . . . . . . . . . 10 {⟨“𝑐𝑣𝑑”⟩} ∈ Fin
1211a1i 11 . . . . . . . . 9 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → {⟨“𝑐𝑣𝑑”⟩} ∈ Fin)
1312ralrimiva 2949 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 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)
1510, 13, 14syl2anc 691 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ∈ Fin)
161nbgrssvtx 40582 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉)
1716ad2antrr 758 . . . . . . . . . . 11 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉)
1817ssdifd 3708 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 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”⟩})
2018, 19syl 17 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2120ralrimiva 2949 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → ∀𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
22 simpr 476 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → 𝑣𝑉)
23 s3iunsndisj 13555 . . . . . . . . 9 (𝑣𝑉Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2422, 23syl 17 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → 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”⟩}))
2621, 24, 25sylc 63 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
277, 15, 26hashiun 14395 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (#‘ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)(#‘ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
284, 27eqtrd 2644 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (#‘(𝑀𝑣)) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)(#‘ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
2928adantr 480 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (#‘(𝑀𝑣)) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)(#‘ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
307, 9syl 17 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin)
3130ad2antrr 758 . . . . . . 7 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ∈ Fin)
3211a1i 11 . . . . . . 7 (((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → {⟨“𝑐𝑣𝑑”⟩} ∈ Fin)
3322adantr 480 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝑣𝑉)
3433anim1i 590 . . . . . . . . 9 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑣𝑉𝑐 ∈ (𝐺 NeighbVtx 𝑣)))
3534ancomd 466 . . . . . . . 8 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣𝑉))
36 s3sndisj 13554 . . . . . . . 8 ((𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣𝑉) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
3735, 36syl 17 . . . . . . 7 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
3831, 32, 37hashiun 14395 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((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)
4139, 40mp1i 13 . . . . . . 7 (((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → (#‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
4241sumeq2dv 14281 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((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))
4531, 43, 44sylancl 693 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Σ𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})1 = ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1))
4638, 42, 453eqtrd 2648 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (#‘ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}) = ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1))
4746sumeq2dv 14281 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)(#‘ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1))
487adantr 480 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((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))
5048, 49sylan 487 . . . . . . . 8 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1))
5150oveq1d 6564 . . . . . . 7 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1) = (((#‘(𝐺 NeighbVtx 𝑣)) − 1) · 1))
521hashnbusgrnn0 40604 . . . . . . . . . 10 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) ∈ ℕ0)
5352nn0red 11229 . . . . . . . . 9 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (#‘(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))
5653, 54, 553syl 18 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (((#‘(𝐺 NeighbVtx 𝑣)) − 1) · 1) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1))
5756ad2antrr 758 . . . . . . 7 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (((#‘(𝐺 NeighbVtx 𝑣)) − 1) · 1) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1))
5851, 57eqtrd 2644 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1) = ((#‘(𝐺 NeighbVtx 𝑣)) − 1))
5958sumeq2dv 14281 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1) = Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘(𝐺 NeighbVtx 𝑣)) − 1))
6052nn0cnd 11230 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) ∈ ℂ)
61 1cnd 9935 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → 1 ∈ ℂ)
6260, 61subcld 10271 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → ((#‘(𝐺 NeighbVtx 𝑣)) − 1) ∈ ℂ)
6362adantr 480 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((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)))
6548, 63, 64syl2anc 691 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((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 )
671hashnbusgrvd 40744 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑣𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
6866, 67sylan 487 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (#‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
69 eqeq1 2614 . . . . . . . . 9 (((VtxDeg‘𝐺)‘𝑣) = (#‘(𝐺 NeighbVtx 𝑣)) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 ↔ (#‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
7069eqcoms 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))
7371, 72oveq12d 6567 . . . . . . . 8 ((#‘(𝐺 NeighbVtx 𝑣)) = 𝐾 → ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
7470, 73syl6bi 242 . . . . . . 7 ((#‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1))))
7568, 74syl 17 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1))))
7675imp 444 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((#‘(𝐺 NeighbVtx 𝑣)) · ((#‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
7759, 65, 763eqtrd 2648 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Σ𝑐 ∈ (𝐺 NeighbVtx 𝑣)((#‘((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) · 1) = (𝐾 · (𝐾 − 1)))
7829, 47, 773eqtrd 2648 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1)))
7978ex 449 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
8079ralrimiva 2949 1 (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  ∪ ciun 4455  Disj wdisj 4553   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℂcc 9813  ℝcr 9814  1c1 9816   · cmul 9820   − cmin 10145  2c2 10947  #chash 12979  Word cword 13146  ⟨“cs3 13438  Σcsu 14264  Vtxcvtx 25673   USGraph cusgr 40379   FinUSGraph cfusgr 40535   NeighbVtx cnbgr 40550  VtxDegcvtxdg 40681   WSPathsN cwwspthsn 41031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-nbgr 40554  df-vtxdg 40682  df-1wlks 40800  df-wlks 40801  df-wlkson 40802  df-trls 40901  df-trlson 40902  df-pths 40923  df-spths 40924  df-pthson 40925  df-spthson 40926  df-wwlks 41033  df-wwlksn 41034  df-wwlksnon 41035  df-wspthsn 41036  df-wspthsnon 41037 This theorem is referenced by:  fusgreghash2wsp  41502
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