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Theorem usgreghash2spotv 26593
 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 ordered triples, we have again k*(k-1) such paths. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Hypothesis
Ref Expression
usgreghash2spot.m 𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})
Assertion
Ref Expression
usgreghash2spotv ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → ∀𝑣𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
Distinct variable groups:   𝑡,𝐸,𝑎   𝑉,𝑎,𝑡   𝐸,𝑎,𝑣,𝑡   𝑣,𝑉,𝑎
Allowed substitution hints:   𝐾(𝑣,𝑡,𝑎)   𝑀(𝑣,𝑡,𝑎)

Proof of Theorem usgreghash2spotv
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgreghash2spot.m . . . . . . . . 9 𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})
21usg2spot2nb 26592 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑣𝑉) → (𝑀𝑣) = 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩})
323expa 1257 . . . . . . 7 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (𝑀𝑣) = 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩})
43fveq2d 6107 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (#‘(𝑀𝑣)) = (#‘ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩}))
5 nbfiusgrafi 25978 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑣𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∈ Fin)
653expa 1257 . . . . . . 7 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∈ Fin)
7 diffi 8077 . . . . . . . . . . 11 ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∈ Fin → ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}) ∈ Fin)
85, 7syl 17 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑣𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}) ∈ Fin)
983expa 1257 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}) ∈ Fin)
109adantr 480 . . . . . . . 8 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}) ∈ Fin)
11 snfi 7923 . . . . . . . . . 10 {⟨𝑐, 𝑣, 𝑑⟩} ∈ Fin
1211a1i 11 . . . . . . . . 9 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) ∧ 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐})) → {⟨𝑐, 𝑣, 𝑑⟩} ∈ Fin)
1312ralrimiva 2949 . . . . . . . 8 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ 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)
1510, 13, 14syl2anc 691 . . . . . . 7 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ 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)
1716ad3antrrr 762 . . . . . . . . . . 11 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → (⟨𝑉, 𝐸⟩ Neighbors 𝑣) ⊆ 𝑉)
1817ssdifd 3708 . . . . . . . . . 10 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ 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⟩})
2018, 19syl 17 . . . . . . . . 9 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩})
2120ralrimiva 2949 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → ∀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 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → 𝑣𝑉)
23 otiunsndisj 4905 . . . . . . . . 9 (𝑣𝑉Disj 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩})
2422, 23syl 17 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → 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⟩}))
2621, 24, 25sylc 63 . . . . . . 7 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → Disj 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩})
276, 15, 26hashiun 14395 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (#‘ 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⟩}))
284, 27eqtrd 2644 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (#‘(𝑀𝑣)) = Σ𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)(#‘ 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩}))
2928adantr 480 . . . 4 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘(𝑀𝑣)) = Σ𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)(#‘ 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩}))
309ad2antrr 758 . . . . . . 7 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}) ∈ Fin)
3111a1i 11 . . . . . . 7 ((((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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) → 𝑐𝑉))
3332ad3antrrr 762 . . . . . . . . 9 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣) → 𝑐𝑉))
3433imp 444 . . . . . . . 8 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → 𝑐𝑉)
3522ad2antrr 758 . . . . . . . 8 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → 𝑣𝑉)
36 otsndisj 4904 . . . . . . . 8 ((𝑐𝑉𝑣𝑉) → Disj 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩})
3734, 35, 36syl2anc 691 . . . . . . 7 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → Disj 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐}){⟨𝑐, 𝑣, 𝑑⟩})
3830, 31, 37hashiun 14395 . . . . . 6 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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)
4139, 40mp1i 13 . . . . . . 7 ((((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) ∧ 𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐})) → (#‘{⟨𝑐, 𝑣, 𝑑⟩}) = 1)
4241sumeq2dv 14281 . . . . . 6 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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))
4530, 43, 44sylancl 693 . . . . . 6 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → Σ𝑑 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐})1 = ((#‘((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐})) · 1))
4638, 42, 453eqtrd 2648 . . . . 5 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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))
4746sumeq2dv 14281 . . . 4 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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))
486adantr 480 . . . . . . . . 9 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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))
5048, 49sylan 487 . . . . . . . 8 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → (#‘((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐})) = ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1))
5150oveq1d 6564 . . . . . . 7 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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)
536, 52syl 17 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) ∈ ℕ0)
5453nn0red 11229 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (#‘(⟨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))
5754, 55, 563syl 18 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1) · 1) = ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1))
5857ad2antrr 758 . . . . . . 7 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → (((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1) · 1) = ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1))
5951, 58eqtrd 2644 . . . . . 6 (((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)) → ((#‘((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐})) · 1) = ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1))
6059sumeq2dv 14281 . . . . 5 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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))
6153nn0cnd 11230 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) ∈ ℂ)
6243a1i 11 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → 1 ∈ ℂ)
6361, 62subcld 10271 . . . . . . 7 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1) ∈ ℂ)
6463adantr 480 . . . . . 6 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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)))
6648, 64, 65syl2anc 691 . . . . 5 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((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 𝐸𝑣𝑉) → (#‘(⟨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))
6968eqcoms 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))
7270, 71oveq12d 6567 . . . . . . . . 9 ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) = 𝐾 → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) · ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
7369, 72syl6bi 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))))
7467, 73syl 17 . . . . . . 7 ((𝑉 USGrph 𝐸𝑣𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) · ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1))))
7574adantlr 747 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) · ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1))))
7675imp 444 . . . . 5 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) · ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
7760, 66, 763eqtrd 2648 . . . 4 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑐 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣)((#‘((⟨𝑉, 𝐸⟩ Neighbors 𝑣) ∖ {𝑐})) · 1) = (𝐾 · (𝐾 − 1)))
7829, 47, 773eqtrd 2648 . . 3 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1)))
7978ex 449 . 2 (((𝑉 USGrph 𝐸𝑉 ∈ Fin) ∧ 𝑣𝑉) → (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
8079ralrimiva 2949 1 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → ∀𝑣𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ⟨cotp 4133  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  ℂcc 9813  ℝcr 9814  1c1 9816   · cmul 9820   − cmin 10145  ℕ0cn0 11169  #chash 12979  Σcsu 14264   USGrph cusg 25859   Neighbors cnbgra 25946   2SPathsOt c2spthot 26383   VDeg cvdg 26420 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-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-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-ot 4134  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-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-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-usgra 25862  df-nbgra 25949  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2wlkonot 26385  df-2spthonot 26387  df-2spthsot 26388  df-vdgr 26421 This theorem is referenced by:  usgreghash2spot  26596
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