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Theorem usgreghash2spot 26596
 Description: In a finite k-regular graph with N vertices there are N times "𝑘 choose 2 " paths with length 2, according to statement 8 in [Huneke] p. 2: "... giving n * ( k 2 ) total paths of length two.", if the direction of traversing the path is not respected. For simple paths of length 2 represented by ordered triples, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
Assertion
Ref Expression
usgreghash2spot ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑉 2SPathsOt 𝐸)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1)))))
Distinct variable groups:   𝑣,𝐸   𝑣,𝐾   𝑣,𝑉

Proof of Theorem usgreghash2spot
Dummy variables 𝑎 𝑠 𝑡 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2676 . . . . . . . . . 10 (𝑠 = 𝑡 → (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ↔ 𝑡 ∈ (𝑉 2SPathsOt 𝐸)))
2 fveq2 6103 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (1st𝑠) = (1st𝑡))
32fveq2d 6107 . . . . . . . . . . 11 (𝑠 = 𝑡 → (2nd ‘(1st𝑠)) = (2nd ‘(1st𝑡)))
43eqeq1d 2612 . . . . . . . . . 10 (𝑠 = 𝑡 → ((2nd ‘(1st𝑠)) = 𝑎 ↔ (2nd ‘(1st𝑡)) = 𝑎))
51, 4anbi12d 743 . . . . . . . . 9 (𝑠 = 𝑡 → ((𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎) ↔ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)))
65cbvrabv 3172 . . . . . . . 8 {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)}
76mpteq2i 4669 . . . . . . 7 (𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)}) = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})
87usgreg2spot 26594 . . . . . 6 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = 𝑦𝑉 ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)))
983adant3 1074 . . . . 5 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = 𝑦𝑉 ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)))
109imp 444 . . . 4 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 2SPathsOt 𝐸) = 𝑦𝑉 ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦))
1110fveq2d 6107 . . 3 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘(𝑉 2SPathsOt 𝐸)) = (#‘ 𝑦𝑉 ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)))
12 simpl 472 . . . . 5 ((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → 𝑉 ∈ Fin)
13 simpr 476 . . . . . . 7 (((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦𝑉) → 𝑦𝑉)
14 3xpfi 8117 . . . . . . . . 9 (𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ Fin)
15 rabexg 4739 . . . . . . . . 9 (((𝑉 × 𝑉) × 𝑉) ∈ Fin → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)} ∈ V)
1614, 15syl 17 . . . . . . . 8 (𝑉 ∈ Fin → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)} ∈ V)
1716ad2antrr 758 . . . . . . 7 (((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦𝑉) → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)} ∈ V)
18 eqeq2 2621 . . . . . . . . . 10 (𝑎 = 𝑦 → ((2nd ‘(1st𝑠)) = 𝑎 ↔ (2nd ‘(1st𝑠)) = 𝑦))
1918anbi2d 736 . . . . . . . . 9 (𝑎 = 𝑦 → ((𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎) ↔ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)))
2019rabbidv 3164 . . . . . . . 8 (𝑎 = 𝑦 → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)} = {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)})
21 eqid 2610 . . . . . . . 8 (𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)}) = (𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})
2220, 21fvmptg 6189 . . . . . . 7 ((𝑦𝑉 ∧ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)} ∈ V) → ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦) = {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)})
2313, 17, 22syl2anc 691 . . . . . 6 (((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦𝑉) → ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦) = {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)})
2414ad2antrr 758 . . . . . . 7 (((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦𝑉) → ((𝑉 × 𝑉) × 𝑉) ∈ Fin)
25 rabfi 8070 . . . . . . 7 (((𝑉 × 𝑉) × 𝑉) ∈ Fin → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)} ∈ Fin)
2624, 25syl 17 . . . . . 6 (((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦𝑉) → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑦)} ∈ Fin)
2723, 26eqeltrd 2688 . . . . 5 (((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦𝑉) → ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦) ∈ Fin)
2872spotmdisj 26595 . . . . . 6 (𝑉 ∈ Fin → Disj 𝑦𝑉 ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦))
2928adantr 480 . . . . 5 ((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Disj 𝑦𝑉 ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦))
3012, 27, 29hashiun 14395 . . . 4 ((𝑉 ∈ Fin ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘ 𝑦𝑉 ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)) = Σ𝑦𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)))
31303ad2antl2 1217 . . 3 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘ 𝑦𝑉 ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)) = Σ𝑦𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)))
327usgreghash2spotv 26593 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → ∀𝑣𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))))
33 ralim 2932 . . . . . . . . 9 (∀𝑣𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ∀𝑣𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))))
3432, 33syl 17 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ∀𝑣𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))))
35343adant3 1074 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ∀𝑣𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))))
3635imp 444 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → ∀𝑣𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1)))
37 fveq2 6103 . . . . . . . . 9 (𝑣 = 𝑦 → ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣) = ((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦))
3837fveq2d 6107 . . . . . . . 8 (𝑣 = 𝑦 → (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)))
3938eqeq1d 2612 . . . . . . 7 (𝑣 = 𝑦 → ((#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1)) ↔ (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)) = (𝐾 · (𝐾 − 1))))
4039rspccva 3281 . . . . . 6 ((∀𝑣𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1)) ∧ 𝑦𝑉) → (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)) = (𝐾 · (𝐾 − 1)))
4136, 40sylan 487 . . . . 5 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦𝑉) → (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)) = (𝐾 · (𝐾 − 1)))
4241sumeq2dv 14281 . . . 4 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑦𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)) = Σ𝑦𝑉 (𝐾 · (𝐾 − 1)))
43 simpl2 1058 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → 𝑉 ∈ Fin)
44 usgfidegfi 26437 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0)
45 r19.26 3046 . . . . . . . . . . 11 (∀𝑣𝑉 (((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ↔ (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾))
46 eleq1 2676 . . . . . . . . . . . . . 14 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0𝐾 ∈ ℕ0))
4746biimpac 502 . . . . . . . . . . . . 13 ((((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → 𝐾 ∈ ℕ0)
4847ralimi 2936 . . . . . . . . . . . 12 (∀𝑣𝑉 (((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → ∀𝑣𝑉 𝐾 ∈ ℕ0)
49 r19.2z 4012 . . . . . . . . . . . . . 14 ((𝑉 ≠ ∅ ∧ ∀𝑣𝑉 𝐾 ∈ ℕ0) → ∃𝑣𝑉 𝐾 ∈ ℕ0)
50 nn0cn 11179 . . . . . . . . . . . . . . . 16 (𝐾 ∈ ℕ0𝐾 ∈ ℂ)
51 kcnktkm1cn 10340 . . . . . . . . . . . . . . . 16 (𝐾 ∈ ℂ → (𝐾 · (𝐾 − 1)) ∈ ℂ)
5250, 51syl 17 . . . . . . . . . . . . . . 15 (𝐾 ∈ ℕ0 → (𝐾 · (𝐾 − 1)) ∈ ℂ)
5352rexlimivw 3011 . . . . . . . . . . . . . 14 (∃𝑣𝑉 𝐾 ∈ ℕ0 → (𝐾 · (𝐾 − 1)) ∈ ℂ)
5449, 53syl 17 . . . . . . . . . . . . 13 ((𝑉 ≠ ∅ ∧ ∀𝑣𝑉 𝐾 ∈ ℕ0) → (𝐾 · (𝐾 − 1)) ∈ ℂ)
5554expcom 450 . . . . . . . . . . . 12 (∀𝑣𝑉 𝐾 ∈ ℕ0 → (𝑉 ≠ ∅ → (𝐾 · (𝐾 − 1)) ∈ ℂ))
5648, 55syl 17 . . . . . . . . . . 11 (∀𝑣𝑉 (((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0 ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 ≠ ∅ → (𝐾 · (𝐾 − 1)) ∈ ℂ))
5745, 56sylbir 224 . . . . . . . . . 10 ((∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0 ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 ≠ ∅ → (𝐾 · (𝐾 − 1)) ∈ ℂ))
5857ex 449 . . . . . . . . 9 (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 ≠ ∅ → (𝐾 · (𝐾 − 1)) ∈ ℂ)))
5958com23 84 . . . . . . . 8 (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0 → (𝑉 ≠ ∅ → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝐾 · (𝐾 − 1)) ∈ ℂ)))
6044, 59syl 17 . . . . . . 7 ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (𝑉 ≠ ∅ → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝐾 · (𝐾 − 1)) ∈ ℂ)))
6160ex 449 . . . . . 6 (𝑉 USGrph 𝐸 → (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝐾 · (𝐾 − 1)) ∈ ℂ))))
62613imp1 1272 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝐾 · (𝐾 − 1)) ∈ ℂ)
63 fsumconst 14364 . . . . 5 ((𝑉 ∈ Fin ∧ (𝐾 · (𝐾 − 1)) ∈ ℂ) → Σ𝑦𝑉 (𝐾 · (𝐾 − 1)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))
6443, 62, 63syl2anc 691 . . . 4 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑦𝑉 (𝐾 · (𝐾 − 1)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))
6542, 64eqtrd 2644 . . 3 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑦𝑉 (#‘((𝑎𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑠)) = 𝑎)})‘𝑦)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))
6611, 31, 653eqtrd 2648 . 2 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘(𝑉 2SPathsOt 𝐸)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))
6766ex 449 1 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑉 2SPathsOt 𝐸)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ∅c0 3874  ∪ 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  1c1 9816   · cmul 9820   − cmin 10145  ℕ0cn0 11169  #chash 12979  Σcsu 14264   USGrph cusg 25859   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:  frgregordn0  26597
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