Theorem fusgreghash2wspv 27199

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 27196. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 10-Jan-2022.)

Hypotheses

Ref	Expression
frgrhash2wsp.v	|- V = (Vtx ` G )
fusgreg2wsp.m	|- M = ( a e. V |-> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } )

Assertion

Ref	Expression
fusgreghash2wspv	|- ( G e. FinUSGraph -> A. v e. V ( ( (VtxDeg ` G ) ` v ) = K -> ( # ` ( M ` v ) ) = ( K x. ( 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 e. V |-> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } )
3	1, 2	fusgr2wsp2nb 27198	. . . . . 6 |- ( ( G e. FinUSGraph /\ v e. V ) -> ( M ` v ) = U_ c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } )
4	3	fveq2d 6195	. . . . 5 |- ( ( G e. FinUSGraph /\ v e. V ) -> ( # ` ( M ` v ) ) = ( # ` U_ c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } ) )
5	4	adantr 481	. . . 4 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) -> ( # ` ( M ` v ) ) = ( # ` U_ c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } ) )
6	1	eleq2i 2693	. . . . . . 7 |- ( v e. V <-> v e. (Vtx ` G ) )
7		nbfiusgrfi 26277	. . . . . . 7 |- ( ( G e. FinUSGraph /\ v e. (Vtx ` G ) ) -> ( G NeighbVtx v ) e. Fin )
8	6, 7	sylan2b 492	. . . . . 6 |- ( ( G e. FinUSGraph /\ v e. V ) -> ( G NeighbVtx v ) e. Fin )
9	8	adantr 481	. . . . 5 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) -> ( G NeighbVtx v ) e. Fin )
10		eqid 2622	. . . . 5 |- ( ( G NeighbVtx v ) \ { c } ) = ( ( G NeighbVtx v ) \ { c } )
11		snfi 8038	. . . . . 6 |- { <" c v d "> } e. Fin
12	11	a1i 11	. . . . 5 |- ( ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) /\ c e. ( G NeighbVtx v ) /\ d e. ( ( G NeighbVtx v ) \ { c } ) ) -> { <" c v d "> } e. Fin )
13	1	nbgrssvtx 26256	. . . . . . . . . . 11 |- ( G e. FinUSGraph -> ( G NeighbVtx v ) C_ V )
14	13	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ c e. ( G NeighbVtx v ) ) -> ( G NeighbVtx v ) C_ V )
15	14	ssdifd 3746	. . . . . . . . 9 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ c e. ( G NeighbVtx v ) ) -> ( ( G NeighbVtx v ) \ { c } ) C_ ( V \ { c } ) )
16		iunss1 4532	. . . . . . . . 9 |- ( ( ( G NeighbVtx v ) \ { c } ) C_ ( V \ { c } ) -> U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } C_ U_ d e. ( V \ { c } ) { <" c v d "> } )
17	15, 16	syl 17	. . . . . . . 8 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ c e. ( G NeighbVtx v ) ) -> U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } C_ U_ d e. ( V \ { c } ) { <" c v d "> } )
18	17	ralrimiva 2966	. . . . . . 7 |- ( ( G e. FinUSGraph /\ v e. V ) -> A. c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } C_ U_ d e. ( V \ { c } ) { <" c v d "> } )
19		simpr 477	. . . . . . . 8 |- ( ( G e. FinUSGraph /\ v e. V ) -> v e. V )
20		s3iunsndisj 13707	. . . . . . . 8 |- ( v e. V -> Disj c e. ( G NeighbVtx v ) U_ d e. ( V \ { c } ) { <" c v d "> } )
21	19, 20	syl 17	. . . . . . 7 |- ( ( G e. FinUSGraph /\ v e. V ) -> Disj c e. ( G NeighbVtx v ) U_ d e. ( V \ { c } ) { <" c v d "> } )
22		disjss2 4623	. . . . . . 7 |- ( A. c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } C_ U_ d e. ( V \ { c } ) { <" c v d "> } -> (Disj c e. ( G NeighbVtx v ) U_ d e. ( V \ { c } ) { <" c v d "> } -> Disj c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } ) )
23	18, 21, 22	sylc 65	. . . . . 6 |- ( ( G e. FinUSGraph /\ v e. V ) -> Disj c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } )
24	23	adantr 481	. . . . 5 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) -> Disj c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } )
25	19	adantr 481	. . . . . . . 8 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) -> v e. V )
26	25	anim1i 592	. . . . . . 7 |- ( ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) /\ c e. ( G NeighbVtx v ) ) -> ( v e. V /\ c e. ( G NeighbVtx v ) ) )
27	26	ancomd 467	. . . . . 6 |- ( ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) /\ c e. ( G NeighbVtx v ) ) -> ( c e. ( G NeighbVtx v ) /\ v e. V ) )
28		s3sndisj 13706	. . . . . 6 |- ( ( c e. ( G NeighbVtx v ) /\ v e. V ) -> Disj d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } )
29	27, 28	syl 17	. . . . 5 |- ( ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) /\ c e. ( G NeighbVtx v ) ) -> Disj d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } )
30		s3cli 13626	. . . . . 6 |- <" c v d "> e. Word _V
31		hashsng 13159	. . . . . 6 |- ( <" c v d "> e. Word _V -> ( # ` { <" c v d "> } ) = 1 )
32	30, 31	mp1i 13	. . . . 5 |- ( ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) /\ c e. ( G NeighbVtx v ) /\ d e. ( ( G NeighbVtx v ) \ { c } ) ) -> ( # ` { <" c v d "> } ) = 1 )
33	9, 10, 12, 24, 29, 32	hash2iun1dif1 14556	. . . 4 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) -> ( # ` U_ c e. ( G NeighbVtx v ) U_ d e. ( ( G NeighbVtx v ) \ { c } ) { <" c v d "> } ) = ( ( # ` ( G NeighbVtx v ) ) x. ( ( # ` ( G NeighbVtx v ) ) - 1 ) ) )
34		fusgrusgr 26214	. . . . . . 7 |- ( G e. FinUSGraph -> G e. USGraph )
35	1	hashnbusgrvd 26424	. . . . . . 7 |- ( ( G e. USGraph /\ v e. V ) -> ( # ` ( G NeighbVtx v ) ) = ( (VtxDeg ` G ) ` v ) )
36	34, 35	sylan 488	. . . . . 6 |- ( ( G e. FinUSGraph /\ v e. V ) -> ( # ` ( G NeighbVtx v ) ) = ( (VtxDeg ` G ) ` v ) )
37		id 22	. . . . . . 7 |- ( ( # ` ( G NeighbVtx v ) ) = ( (VtxDeg ` G ) ` v ) -> ( # ` ( G NeighbVtx v ) ) = ( (VtxDeg ` G ) ` v ) )
38		oveq1 6657	. . . . . . 7 |- ( ( # ` ( G NeighbVtx v ) ) = ( (VtxDeg ` G ) ` v ) -> ( ( # ` ( G NeighbVtx v ) ) - 1 ) = ( ( (VtxDeg ` G ) ` v ) - 1 ) )
39	37, 38	oveq12d 6668	. . . . . 6 |- ( ( # ` ( G NeighbVtx v ) ) = ( (VtxDeg ` G ) ` v ) -> ( ( # ` ( G NeighbVtx v ) ) x. ( ( # ` ( G NeighbVtx v ) ) - 1 ) ) = ( ( (VtxDeg ` G ) ` v ) x. ( ( (VtxDeg ` G ) ` v ) - 1 ) ) )
40	36, 39	syl 17	. . . . 5 |- ( ( G e. FinUSGraph /\ v e. V ) -> ( ( # ` ( G NeighbVtx v ) ) x. ( ( # ` ( G NeighbVtx v ) ) - 1 ) ) = ( ( (VtxDeg ` G ) ` v ) x. ( ( (VtxDeg ` G ) ` v ) - 1 ) ) )
41		id 22	. . . . . 6 |- ( ( (VtxDeg ` G ) ` v ) = K -> ( (VtxDeg ` G ) ` v ) = K )
42		oveq1 6657	. . . . . 6 |- ( ( (VtxDeg ` G ) ` v ) = K -> ( ( (VtxDeg ` G ) ` v ) - 1 ) = ( K - 1 ) )
43	41, 42	oveq12d 6668	. . . . 5 |- ( ( (VtxDeg ` G ) ` v ) = K -> ( ( (VtxDeg ` G ) ` v ) x. ( ( (VtxDeg ` G ) ` v ) - 1 ) ) = ( K x. ( K - 1 ) ) )
44	40, 43	sylan9eq 2676	. . . 4 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) -> ( ( # ` ( G NeighbVtx v ) ) x. ( ( # ` ( G NeighbVtx v ) ) - 1 ) ) = ( K x. ( K - 1 ) ) )
45	5, 33, 44	3eqtrd 2660	. . 3 |- ( ( ( G e. FinUSGraph /\ v e. V ) /\ ( (VtxDeg ` G ) ` v ) = K ) -> ( # ` ( M ` v ) ) = ( K x. ( K - 1 ) ) )
46	45	ex 450	. 2 |- ( ( G e. FinUSGraph /\ v e. V ) -> ( ( (VtxDeg ` G ) ` v ) = K -> ( # ` ( M ` v ) ) = ( K x. ( K - 1 ) ) ) )
47	46	ralrimiva 2966	1 |- ( G e. FinUSGraph -> A. v e. V ( ( (VtxDeg ` G ) ` v ) = K -> ( # ` ( M ` v ) ) = ( K x. ( K - 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   U_ciun 4520  Disj wdisj 4620   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   x. cmul 9941   - cmin 10266   2c2 11070   #chash 13117  Word cword 13291   <"cs3 13587  Vtxcvtx 25874   USGraph cusgr 26044   FinUSGraph cfusgr 26208   NeighbVtx cnbgr 26224  VtxDegcvtxdg 26361   WSPathsN cwwspthsn 26720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-nbgr 26228  df-vtxdg 26362  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsn 26725  df-wspthsnon 26726

This theorem is referenced by:  fusgreghash2wsp  27202