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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 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((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.

Step	Hyp	Ref	Expression
1		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ↔ 𝑡 ∈ (𝑉 2SPathsOt 𝐸)))
2		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (1^st ‘𝑠) = (1^st ‘𝑡))
3	2	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (2^nd ‘(1^st ‘𝑠)) = (2^nd ‘(1^st ‘𝑡)))
4	3	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((2^nd ‘(1^st ‘𝑠)) = 𝑎 ↔ (2^nd ‘(1^st ‘𝑡)) = 𝑎))
5	1, 4	anbi12d 743	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ((𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎) ↔ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑡)) = 𝑎)))
6	5	cbvrabv 3172	. . . . . . . 8 ⊢ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑡)) = 𝑎)}
7	6	mpteq2i 4669	. . . . . . 7 ⊢ (𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)}) = (𝑎 ∈ 𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑡)) = 𝑎)})
8	7	usgreg2spot 26594	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = ∪ 𝑦 ∈ 𝑉 ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)))
9	8	3adant3 1074	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = ∪ 𝑦 ∈ 𝑉 ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)))
10	9	imp 444	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 2SPathsOt 𝐸) = ∪ 𝑦 ∈ 𝑉 ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦))
11	10	fveq2d 6107	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘(𝑉 2SPathsOt 𝐸)) = (#‘∪ 𝑦 ∈ 𝑉 ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)))
12		simpl 472	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → 𝑉 ∈ Fin)
13		simpr 476	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
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) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)} ∈ V)
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)} ∈ V)
17	16	ad2antrr 758	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦 ∈ 𝑉) → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)} ∈ V)
18		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → ((2^nd ‘(1^st ‘𝑠)) = 𝑎 ↔ (2^nd ‘(1^st ‘𝑠)) = 𝑦))
19	18	anbi2d 736	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → ((𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎) ↔ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)))
20	19	rabbidv 3164	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)} = {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)})
21		eqid 2610	. . . . . . . 8 ⊢ (𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)}) = (𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})
22	20, 21	fvmptg 6189	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑉 ∧ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)} ∈ V) → ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦) = {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)})
23	13, 17, 22	syl2anc 691	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦 ∈ 𝑉) → ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦) = {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)})
24	14	ad2antrr 758	. . . . . . 7 ⊢ (((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦 ∈ 𝑉) → ((𝑉 × 𝑉) × 𝑉) ∈ Fin)
25		rabfi 8070	. . . . . . 7 ⊢ (((𝑉 × 𝑉) × 𝑉) ∈ Fin → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)} ∈ Fin)
26	24, 25	syl 17	. . . . . 6 ⊢ (((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦 ∈ 𝑉) → {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑦)} ∈ Fin)
27	23, 26	eqeltrd 2688	. . . . 5 ⊢ (((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦 ∈ 𝑉) → ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦) ∈ Fin)
28	7	2spotmdisj 26595	. . . . . 6 ⊢ (𝑉 ∈ Fin → Disj 𝑦 ∈ 𝑉 ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦))
29	28	adantr 480	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Disj 𝑦 ∈ 𝑉 ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦))
30	12, 27, 29	hashiun 14395	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘∪ 𝑦 ∈ 𝑉 ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)) = Σ𝑦 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)))
31	30	3ad2antl2 1217	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘∪ 𝑦 ∈ 𝑉 ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)) = Σ𝑦 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)))
32	7	usgreghash2spotv 26593	. . . . . . . . 9 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → ∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))))
33		ralim 2932	. . . . . . . . 9 ⊢ (∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))))
34	32, 33	syl 17	. . . . . . . 8 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))))
35	34	3adant3 1074	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1))))
36	35	imp 444	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → ∀𝑣 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1)))
37		fveq2 6103	. . . . . . . . 9 ⊢ (𝑣 = 𝑦 → ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣) = ((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦))
38	37	fveq2d 6107	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)))
39	38	eqeq1d 2612	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ((#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1)) ↔ (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)) = (𝐾 · (𝐾 − 1))))
40	39	rspccva 3281	. . . . . 6 ⊢ ((∀𝑣 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑣)) = (𝐾 · (𝐾 − 1)) ∧ 𝑦 ∈ 𝑉) → (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)) = (𝐾 · (𝐾 − 1)))
41	36, 40	sylan 487	. . . . 5 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑦 ∈ 𝑉) → (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)) = (𝐾 · (𝐾 − 1)))
42	41	sumeq2dv 14281	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑦 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)) = Σ𝑦 ∈ 𝑉 (𝐾 · (𝐾 − 1)))
43		simpl2 1058	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → 𝑉 ∈ Fin)
44		usgfidegfi 26437	. . . . . . . 8 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀)
45		r19.26 3046	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ↔ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾))
46		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ ↔ 𝐾 ∈ ℕ₀))
47	46	biimpac 502	. . . . . . . . . . . . 13 ⊢ ((((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → 𝐾 ∈ ℕ₀)
48	47	ralimi 2936	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → ∀𝑣 ∈ 𝑉 𝐾 ∈ ℕ₀)
49		r19.2z 4012	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ≠ ∅ ∧ ∀𝑣 ∈ 𝑉 𝐾 ∈ ℕ₀) → ∃𝑣 ∈ 𝑉 𝐾 ∈ ℕ₀)
50		nn0cn 11179	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ₀ → 𝐾 ∈ ℂ)
51		kcnktkm1cn 10340	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℂ → (𝐾 · (𝐾 − 1)) ∈ ℂ)
52	50, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 · (𝐾 − 1)) ∈ ℂ)
53	52	rexlimivw 3011	. . . . . . . . . . . . . 14 ⊢ (∃𝑣 ∈ 𝑉 𝐾 ∈ ℕ₀ → (𝐾 · (𝐾 − 1)) ∈ ℂ)
54	49, 53	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑉 ≠ ∅ ∧ ∀𝑣 ∈ 𝑉 𝐾 ∈ ℕ₀) → (𝐾 · (𝐾 − 1)) ∈ ℂ)
55	54	expcom 450	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ 𝑉 𝐾 ∈ ℕ₀ → (𝑉 ≠ ∅ → (𝐾 · (𝐾 − 1)) ∈ ℂ))
56	48, 55	syl 17	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ 𝑉 (((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ ∧ ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 ≠ ∅ → (𝐾 · (𝐾 − 1)) ∈ ℂ))
57	45, 56	sylbir 224	. . . . . . . . . 10 ⊢ ((∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 ≠ ∅ → (𝐾 · (𝐾 − 1)) ∈ ℂ))
58	57	ex 449	. . . . . . . . 9 ⊢ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 ≠ ∅ → (𝐾 · (𝐾 − 1)) ∈ ℂ)))
59	58	com23 84	. . . . . . . 8 ⊢ (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ₀ → (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝐾 · (𝐾 − 1)) ∈ ℂ)))
60	44, 59	syl 17	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝐾 · (𝐾 − 1)) ∈ ℂ)))
61	60	ex 449	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ Fin → (𝑉 ≠ ∅ → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝐾 · (𝐾 − 1)) ∈ ℂ))))
62	61	3imp1 1272	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝐾 · (𝐾 − 1)) ∈ ℂ)
63		fsumconst 14364	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ (𝐾 · (𝐾 − 1)) ∈ ℂ) → Σ𝑦 ∈ 𝑉 (𝐾 · (𝐾 − 1)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))
64	43, 62, 63	syl2anc 691	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑦 ∈ 𝑉 (𝐾 · (𝐾 − 1)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))
65	42, 64	eqtrd 2644	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → Σ𝑦 ∈ 𝑉 (#‘((𝑎 ∈ 𝑉 ↦ {𝑠 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑠 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2^nd ‘(1^st ‘𝑠)) = 𝑎)})‘𝑦)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))
66	11, 31, 65	3eqtrd 2648	. 2 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (#‘(𝑉 2SPathsOt 𝐸)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))
67	66	ex 449	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((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 1^st c1st 7057 2^nd 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

W3C validator