Theorem fusgreghash2wsp 27202

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 length 3 strings, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 19-May-2021.) (Proof shortened by AV, 12-Jan-2022.)

Hypothesis

Ref	Expression
fusgreghash2wsp.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
fusgreghash2wsp	|- ( ( G e. FinUSGraph /\ V =/= (/) ) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( # ` ( 2 WSPathsN G ) ) = ( ( # ` V ) x. ( K x. ( K - 1 ) ) ) ) )

Distinct variable groups:   v, G   v, K   v, V

Proof of Theorem fusgreghash2wsp

Dummy variables a s t y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fusgreghash2wsp.v	. . . . . 6 |- V = (Vtx ` G )
2		fveq1 6190	. . . . . . . . 9 |- ( s = t -> ( s ` 1 ) = ( t ` 1 ) )
3	2	eqeq1d 2624	. . . . . . . 8 |- ( s = t -> ( ( s ` 1 ) = a <-> ( t ` 1 ) = a ) )
4	3	cbvrabv 3199	. . . . . . 7 |- { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } = { t e. ( 2 WSPathsN G ) | ( t ` 1 ) = a }
5	4	mpteq2i 4741	. . . . . 6 |- ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) = ( a e. V |-> { t e. ( 2 WSPathsN G ) | ( t ` 1 ) = a } )
6	1, 5	fusgreg2wsp 27200	. . . . 5 |- ( G e. FinUSGraph -> ( 2 WSPathsN G ) = U_ y e. V ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) )
7	6	ad2antrr 762	. . . 4 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> ( 2 WSPathsN G ) = U_ y e. V ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) )
8	7	fveq2d 6195	. . 3 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> ( # ` ( 2 WSPathsN G ) ) = ( # ` U_ y e. V ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) )
9	1	fusgrvtxfi 26211	. . . . 5 |- ( G e. FinUSGraph -> V e. Fin )
10		eqeq2 2633	. . . . . . . . 9 |- ( a = y -> ( ( s ` 1 ) = a <-> ( s ` 1 ) = y ) )
11	10	rabbidv 3189	. . . . . . . 8 |- ( a = y -> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } = { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = y } )
12		eqid 2622	. . . . . . . 8 |- ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) = ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } )
13		ovex 6678	. . . . . . . . 9 |- ( 2 WSPathsN G ) e. _V
14	13	rabex 4813	. . . . . . . 8 |- { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = y } e. _V
15	11, 12, 14	fvmpt 6282	. . . . . . 7 |- ( y e. V -> ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) = { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = y } )
16	15	adantl 482	. . . . . 6 |- ( ( G e. FinUSGraph /\ y e. V ) -> ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) = { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = y } )
17		eqid 2622	. . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
18	17	fusgrvtxfi 26211	. . . . . . . 8 |- ( G e. FinUSGraph -> (Vtx ` G ) e. Fin )
19		wspthnfi 26815	. . . . . . . 8 |- ( (Vtx ` G ) e. Fin -> ( 2 WSPathsN G ) e. Fin )
20		rabfi 8185	. . . . . . . 8 |- ( ( 2 WSPathsN G ) e. Fin -> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = y } e. Fin )
21	18, 19, 20	3syl 18	. . . . . . 7 |- ( G e. FinUSGraph -> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = y } e. Fin )
22	21	adantr 481	. . . . . 6 |- ( ( G e. FinUSGraph /\ y e. V ) -> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = y } e. Fin )
23	16, 22	eqeltrd 2701	. . . . 5 |- ( ( G e. FinUSGraph /\ y e. V ) -> ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) e. Fin )
24	1, 5	2wspmdisj 27201	. . . . . 6 |- Disj y e. V ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y )
25	24	a1i 11	. . . . 5 |- ( G e. FinUSGraph -> Disj y e. V ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) )
26	9, 23, 25	hashiun 14554	. . . 4 |- ( G e. FinUSGraph -> ( # ` U_ y e. V ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) = sum_ y e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) )
27	26	ad2antrr 762	. . 3 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> ( # ` U_ y e. V ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) = sum_ y e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) )
28	1, 5	fusgreghash2wspv 27199	. . . . . . . . 9 |- ( G e. FinUSGraph -> A. v e. V ( ( (VtxDeg ` G ) ` v ) = K -> ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( K x. ( K - 1 ) ) ) )
29		ralim 2948	. . . . . . . . 9 |- ( A. v e. V ( ( (VtxDeg ` G ) ` v ) = K -> ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( K x. ( K - 1 ) ) ) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> A. v e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( K x. ( K - 1 ) ) ) )
30	28, 29	syl 17	. . . . . . . 8 |- ( G e. FinUSGraph -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> A. v e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( K x. ( K - 1 ) ) ) )
31	30	adantr 481	. . . . . . 7 |- ( ( G e. FinUSGraph /\ V =/= (/) ) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> A. v e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( K x. ( K - 1 ) ) ) )
32	31	imp 445	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> A. v e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( K x. ( K - 1 ) ) )
33		fveq2 6191	. . . . . . . . 9 |- ( v = y -> ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) = ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) )
34	33	fveq2d 6195	. . . . . . . 8 |- ( v = y -> ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) )
35	34	eqeq1d 2624	. . . . . . 7 |- ( v = y -> ( ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( K x. ( K - 1 ) ) <-> ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) = ( K x. ( K - 1 ) ) ) )
36	35	rspccva 3308	. . . . . 6 |- ( ( A. v e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` v ) ) = ( K x. ( K - 1 ) ) /\ y e. V ) -> ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) = ( K x. ( K - 1 ) ) )
37	32, 36	sylan 488	. . . . 5 |- ( ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) /\ y e. V ) -> ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) = ( K x. ( K - 1 ) ) )
38	37	sumeq2dv 14433	. . . 4 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> sum_ y e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) = sum_ y e. V ( K x. ( K - 1 ) ) )
39	9	adantr 481	. . . . 5 |- ( ( G e. FinUSGraph /\ V =/= (/) ) -> V e. Fin )
40		eqid 2622	. . . . . . . . 9 |- (VtxDeg ` G ) = (VtxDeg ` G )
41	1, 40	fusgrregdegfi 26465	. . . . . . . 8 |- ( ( G e. FinUSGraph /\ V =/= (/) ) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> K e. NN0 ) )
42	41	imp 445	. . . . . . 7 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> K e. NN0 )
43	42	nn0cnd 11353	. . . . . 6 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> K e. CC )
44		kcnktkm1cn 10461	. . . . . 6 |- ( K e. CC -> ( K x. ( K - 1 ) ) e. CC )
45	43, 44	syl 17	. . . . 5 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> ( K x. ( K - 1 ) ) e. CC )
46		fsumconst 14522	. . . . 5 |- ( ( V e. Fin /\ ( K x. ( K - 1 ) ) e. CC ) -> sum_ y e. V ( K x. ( K - 1 ) ) = ( ( # ` V ) x. ( K x. ( K - 1 ) ) ) )
47	39, 45, 46	syl2an2r 876	. . . 4 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> sum_ y e. V ( K x. ( K - 1 ) ) = ( ( # ` V ) x. ( K x. ( K - 1 ) ) ) )
48	38, 47	eqtrd 2656	. . 3 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> sum_ y e. V ( # ` ( ( a e. V |-> { s e. ( 2 WSPathsN G ) | ( s ` 1 ) = a } ) ` y ) ) = ( ( # ` V ) x. ( K x. ( K - 1 ) ) ) )
49	8, 27, 48	3eqtrd 2660	. 2 |- ( ( ( G e. FinUSGraph /\ V =/= (/) ) /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> ( # ` ( 2 WSPathsN G ) ) = ( ( # ` V ) x. ( K x. ( K - 1 ) ) ) )
50	49	ex 450	1 |- ( ( G e. FinUSGraph /\ V =/= (/) ) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( # ` ( 2 WSPathsN G ) ) = ( ( # ` V ) x. ( K x. ( K - 1 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   (/)c0 3915   U_ciun 4520  Disj wdisj 4620   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   1c1 9937   x. cmul 9941   - cmin 10266   2c2 11070   NN0cn0 11292   #chash 13117   sum_csu 14416  Vtxcvtx 25874   FinUSGraph cfusgr 26208  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:  frrusgrord0  27204