Theorem fusgreghash2wspv 30823

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 directed simple paths of length 2 represented by length 3 strings, we have again k*(k-1) such paths, see also comment of frgrhash2wsp 30820. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 12-Feb-2022.)

Hypotheses

Ref	Expression
frgrhash2wsp.v	⊢ 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m	⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})

Assertion

Ref	Expression
fusgreghash2wspv	⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1))))

Distinct variable groups:   𝐺,𝑎   𝑉,𝑎   𝑤,𝐺,𝑎,𝑣

Allowed substitution hints:   𝐾(𝑤, 𝑣, 𝑎)   𝑀(𝑤, 𝑣, 𝑎)   𝑉(𝑤, 𝑣)

Proof of Theorem fusgreghash2wspv

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrhash2wsp.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
2		fusgreg2wsp.m	. . . . . . 7 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
3	1, 2	fusgr2wsp2nb 30822	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (𝑀‘𝑣) = ∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
4	3	fveq2d 6886	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝑀‘𝑣)) = (♯‘∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
5	4	adantr 486	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀‘𝑣)) = (♯‘∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
6	1	eleq2i 2854	. . . . . . 7 ⊢ (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtx‘𝐺))
7		nbfiusgrfi 29843	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
8	6, 7	sylan2b 606	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
9	8	adantr 486	. . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
10		eqid 2762	. . . . 5 ⊢ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) = ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})
11		snfi 9054	. . . . . 6 ⊢ {⟨“𝑐𝑣𝑑”⟩} ∈ Fin
12	11	a1i 11	. . . . 5 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → {⟨“𝑐𝑣𝑑”⟩} ∈ Fin)
13	1	nbgrssvtx 29810	. . . . . . . . . . 11 ⊢ (𝐺 NeighbVtx 𝑣) ⊆ 𝑉
14	13	a1i 11	. . . . . . . . . 10 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉)
15	14	ssdifd 4095	. . . . . . . . 9 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}))
16		iunss1 4969	. . . . . . . . 9 ⊢ (((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}) → ∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
17	15, 16	syl 18	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
18	17	ralrimiva 3156	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → ∀𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
19		simpr 490	. . . . . . . 8 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
20		s3iunsndisj 15045	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑉 → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
21	19, 20	syl 18	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
22		disjss2 5077	. . . . . . 7 ⊢ (∀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”⟩}))
23	18, 21, 22	sylc 66	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
24	23	adantr 486	. . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
25	19	adantr 486	. . . . . . 7 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝑣 ∈ 𝑉)
26	25	anim1ci 628	. . . . . 6 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉))
27		s3sndisj 15044	. . . . . 6 ⊢ ((𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
28	26, 27	syl 18	. . . . 5 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
29		s3cli 14956	. . . . . 6 ⊢ ⟨“𝑐𝑣𝑑”⟩ ∈ Word V
30		hashsng 14437	. . . . . 6 ⊢ (⟨“𝑐𝑣𝑑”⟩ ∈ Word V → (♯‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
31	29, 30	mp1i 14	. . . . 5 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → (♯‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
32	9, 10, 12, 24, 28, 31	hash2iun1dif1 15915	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}) = ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)))
33		fusgrusgr 29790	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
34	1	hashnbusgrvd 29996	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
35	33, 34	sylan 592	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
36		id 23	. . . . . . 7 ⊢ ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
37		oveq1 7424	. . . . . . 7 ⊢ ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) − 1) = (((VtxDeg‘𝐺)‘𝑣) − 1))
38	36, 37	oveq12d 7435	. . . . . 6 ⊢ ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)))
39	35, 38	syl 18	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)))
40		id 23	. . . . . 6 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
41		oveq1 7424	. . . . . 6 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) − 1) = (𝐾 − 1))
42	40, 41	oveq12d 7435	. . . . 5 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)) = (𝐾 · (𝐾 − 1)))
43	39, 42	sylan9eq 2817	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
44	5, 32, 43	3eqtrd 2801	. . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1)))
45	44	ex 418	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1))))
46	45	ralrimiva 3156	1 ⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  {csn 4587  ∪ ciun 4954  Disj wdisj 5074   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417  Fincfn 8956  1c1 11129   · cmul 11133   − cmin 11469  2c2 12323  ♯chash 14398  Word cword 14582  ⟨“cs3 14917  Vtxcvtx 29461  USGraphcusgr 29617  FinUSGraphcfusgr 29784   NeighbVtx cnbgr 29800  VtxDegcvtxdg 29933   WSPathsN cwwspthsn 30304

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-disj 5075  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-oi 9486  df-dju 9910  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-n0 12533  df-xnn0 12606  df-z 12620  df-uz 12892  df-rp 13047  df-xadd 13168  df-fz 13566  df-fzo 13714  df-seq 14070  df-exp 14130  df-hash 14399  df-word 14583  df-concat 14640  df-s1 14667  df-s2 14923  df-s3 14924  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-clim 15579  df-sum 15778  df-vtx 29463  df-iedg 29464  df-edg 29513  df-uhgr 29523  df-ushgr 29524  df-upgr 29547  df-umgr 29548  df-uspgr 29618  df-usgr 29619  df-fusgr 29785  df-nbgr 29801  df-vtxdg 29934  df-wlks 30067  df-wlkson 30068  df-trls 30162  df-trlson 30163  df-pths 30186  df-spths 30187  df-pthson 30188  df-spthson 30189  df-wwlks 30306  df-wwlksn 30307  df-wwlksnon 30308  df-wspthsn 30309  df-wspthsnon 30310

This theorem is used by:  fusgreghash2wsp  30826