MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgreghash2spotv Structured version   Unicode version

Theorem usgreghash2spotv 25042
Description: According to the proof of the third claim in the proof of the friendship theorem 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 simple paths of length 2 represented by ordered triples, we have again k*(k-1) such paths. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Hypothesis
Ref Expression
usgreghash2spot.m  |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) } )
Assertion
Ref Expression
usgreghash2spotv  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( M `  v )
)  =  ( K  x.  ( K  - 
1 ) ) ) )
Distinct variable groups:    t, E, a    V, a, t    E, a, v, t    v, V, a
Allowed substitution hints:    K( v, t, a)    M( v, t, a)

Proof of Theorem usgreghash2spotv
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgreghash2spot.m . . . . . . . . 9  |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) } )
21usg2spot2nb 25041 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  v  e.  V )  ->  ( M `  v )  =  U_ c  e.  (
<. V ,  E >. Neighbors  v
) U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. } )
323expa 1197 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( M `  v )  =  U_ c  e.  ( <. V ,  E >. Neighbors  v )
U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. } )
43fveq2d 5860 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( M `  v )
)  =  ( # `  U_ c  e.  (
<. V ,  E >. Neighbors  v
) U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. } ) )
5 nbfiusgrafi 24425 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  v  e.  V )  ->  ( <. V ,  E >. Neighbors  v
)  e.  Fin )
653expa 1197 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( <. V ,  E >. Neighbors  v )  e.  Fin )
7 diffi 7753 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. Neighbors  v
)  e.  Fin  ->  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } )  e. 
Fin )
85, 7syl 16 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  v  e.  V )  ->  (
( <. V ,  E >. Neighbors 
v )  \  {
c } )  e. 
Fin )
983expa 1197 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( ( <. V ,  E >. Neighbors  v
)  \  { c } )  e.  Fin )
109adantr 465 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  -> 
( ( <. V ,  E >. Neighbors  v )  \  {
c } )  e. 
Fin )
11 snfi 7598 . . . . . . . . . 10  |-  { <. c ,  v ,  d
>. }  e.  Fin
1211a1i 11 . . . . . . . . 9  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  /\  d  e.  ( ( <. V ,  E >. Neighbors  v
)  \  { c } ) )  ->  { <. c ,  v ,  d >. }  e.  Fin )
1312ralrimiva 2857 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  ->  A. d  e.  (
( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. }  e.  Fin )
14 iunfi 7810 . . . . . . . 8  |-  ( ( ( ( <. V ,  E >. Neighbors  v )  \  {
c } )  e. 
Fin  /\  A. d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) {
<. c ,  v ,  d >. }  e.  Fin )  ->  U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. }  e.  Fin )
1510, 13, 14syl2anc 661 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  ->  U_ d  e.  (
( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. }  e.  Fin )
16 nbgrassvt 24409 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  v )  C_  V
)
1716ad3antrrr 729 . . . . . . . . . . 11  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  -> 
( <. V ,  E >. Neighbors 
v )  C_  V
)
1817ssdifd 3625 . . . . . . . . . 10  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  -> 
( ( <. V ,  E >. Neighbors  v )  \  {
c } )  C_  ( V  \  { c } ) )
19 iunss1 4327 . . . . . . . . . 10  |-  ( ( ( <. V ,  E >. Neighbors 
v )  \  {
c } )  C_  ( V  \  { c } )  ->  U_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) {
<. c ,  v ,  d >. }  C_  U_ d  e.  ( V  \  {
c } ) {
<. c ,  v ,  d >. } )
2018, 19syl 16 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  ->  U_ d  e.  (
( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. }  C_  U_ d  e.  ( V  \  {
c } ) {
<. c ,  v ,  d >. } )
2120ralrimiva 2857 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  A. c  e.  ( <. V ,  E >. Neighbors 
v ) U_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) {
<. c ,  v ,  d >. }  C_  U_ d  e.  ( V  \  {
c } ) {
<. c ,  v ,  d >. } )
22 simpr 461 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  v  e.  V )
23 otiunsndisj 4743 . . . . . . . . 9  |-  ( v  e.  V  -> Disj  c  e.  ( <. V ,  E >. Neighbors 
v ) U_ d  e.  ( V  \  {
c } ) {
<. c ,  v ,  d >. } )
2422, 23syl 16 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  -> Disj  c  e.  (
<. V ,  E >. Neighbors  v
) U_ d  e.  ( V  \  { c } ) { <. c ,  v ,  d
>. } )
25 disjss2 4410 . . . . . . . 8  |-  ( A. c  e.  ( <. V ,  E >. Neighbors  v )
U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. }  C_  U_ d  e.  ( V  \  {
c } ) {
<. c ,  v ,  d >. }  ->  (Disj  c  e.  ( <. V ,  E >. Neighbors  v ) U_ d  e.  ( V  \  {
c } ) {
<. c ,  v ,  d >. }  -> Disj  c  e.  ( <. V ,  E >. Neighbors 
v ) U_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) {
<. c ,  v ,  d >. } ) )
2621, 24, 25sylc 60 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  -> Disj  c  e.  (
<. V ,  E >. Neighbors  v
) U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. } )
276, 15, 26hashiun 13617 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  U_ c  e.  ( <. V ,  E >. Neighbors 
v ) U_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) {
<. c ,  v ,  d >. } )  = 
sum_ c  e.  (
<. V ,  E >. Neighbors  v
) ( # `  U_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) {
<. c ,  v ,  d >. } ) )
284, 27eqtrd 2484 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( M `  v )
)  =  sum_ c  e.  ( <. V ,  E >. Neighbors 
v ) ( # `  U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. } ) )
2928adantr 465 . . . 4  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( # `  ( M `  v )
)  =  sum_ c  e.  ( <. V ,  E >. Neighbors 
v ) ( # `  U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. } ) )
309ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  (
( <. V ,  E >. Neighbors 
v )  \  {
c } )  e. 
Fin )
3111a1i 11 . . . . . . 7  |-  ( ( ( ( ( ( V USGrph  E  /\  V  e. 
Fin )  /\  v  e.  V )  /\  (
( V VDeg  E ) `  v )  =  K )  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  /\  d  e.  ( ( <. V ,  E >. Neighbors  v
)  \  { c } ) )  ->  { <. c ,  v ,  d >. }  e.  Fin )
32 nbgraisvtx 24407 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( c  e.  ( <. V ,  E >. Neighbors 
v )  ->  c  e.  V ) )
3332ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( c  e.  ( <. V ,  E >. Neighbors 
v )  ->  c  e.  V ) )
3433imp 429 . . . . . . . 8  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  c  e.  V )
3522ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  v  e.  V )
36 otsndisj 4742 . . . . . . . 8  |-  ( ( c  e.  V  /\  v  e.  V )  -> Disj  d  e.  ( ( <. V ,  E >. Neighbors  v
)  \  { c } ) { <. c ,  v ,  d
>. } )
3734, 35, 36syl2anc 661 . . . . . . 7  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  -> Disj  d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) {
<. c ,  v ,  d >. } )
3830, 31, 37hashiun 13617 . . . . . 6  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  ( # `
 U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. } )  = 
sum_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) (
# `  { <. c ,  v ,  d
>. } ) )
39 otex 4702 . . . . . . . 8  |-  <. c ,  v ,  d
>.  e.  _V
40 hashsng 12419 . . . . . . . 8  |-  ( <.
c ,  v ,  d >.  e.  _V  ->  ( # `  { <. c ,  v ,  d >. } )  =  1 )
4139, 40mp1i 12 . . . . . . 7  |-  ( ( ( ( ( ( V USGrph  E  /\  V  e. 
Fin )  /\  v  e.  V )  /\  (
( V VDeg  E ) `  v )  =  K )  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  /\  d  e.  ( ( <. V ,  E >. Neighbors  v
)  \  { c } ) )  -> 
( # `  { <. c ,  v ,  d
>. } )  =  1 )
4241sumeq2dv 13506 . . . . . 6  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  sum_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) (
# `  { <. c ,  v ,  d
>. } )  =  sum_ d  e.  ( ( <. V ,  E >. Neighbors  v
)  \  { c } ) 1 )
43 ax-1cn 9553 . . . . . . 7  |-  1  e.  CC
44 fsumconst 13586 . . . . . . 7  |-  ( ( ( ( <. V ,  E >. Neighbors  v )  \  {
c } )  e. 
Fin  /\  1  e.  CC )  ->  sum_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) 1  =  ( ( # `  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) )  x.  1 ) )
4530, 43, 44sylancl 662 . . . . . 6  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  sum_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) 1  =  ( ( # `  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) )  x.  1 ) )
4638, 42, 453eqtrd 2488 . . . . 5  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  ( # `
 U_ d  e.  ( ( <. V ,  E >. Neighbors 
v )  \  {
c } ) {
<. c ,  v ,  d >. } )  =  ( ( # `  (
( <. V ,  E >. Neighbors 
v )  \  {
c } ) )  x.  1 ) )
4746sumeq2dv 13506 . . . 4  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  sum_ c  e.  (
<. V ,  E >. Neighbors  v
) ( # `  U_ d  e.  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) {
<. c ,  v ,  d >. } )  = 
sum_ c  e.  (
<. V ,  E >. Neighbors  v
) ( ( # `  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) )  x.  1 ) )
486adantr 465 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( <. V ,  E >. Neighbors  v )  e.  Fin )
49 hashdifsn 12458 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. Neighbors 
v )  e.  Fin  /\  c  e.  ( <. V ,  E >. Neighbors  v
) )  ->  ( # `
 ( ( <. V ,  E >. Neighbors  v
)  \  { c } ) )  =  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) )
5048, 49sylan 471 . . . . . . . 8  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  ( # `
 ( ( <. V ,  E >. Neighbors  v
)  \  { c } ) )  =  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) )
5150oveq1d 6296 . . . . . . 7  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  (
( # `  ( (
<. V ,  E >. Neighbors  v
)  \  { c } ) )  x.  1 )  =  ( ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  x.  1 ) )
52 hashcl 12409 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. Neighbors  v
)  e.  Fin  ->  (
# `  ( <. V ,  E >. Neighbors  v ) )  e.  NN0 )
536, 52syl 16 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  e.  NN0 )
5453nn0red 10860 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  e.  RR )
55 peano2rem 9891 . . . . . . . . 9  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  e.  RR  ->  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  e.  RR )
56 ax-1rid 9565 . . . . . . . . 9  |-  ( ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  e.  RR  ->  ( ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  x.  1 )  =  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 ) )
5754, 55, 563syl 20 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( (
( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  x.  1 )  =  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 ) )
5857ad2antrr 725 . . . . . . 7  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  (
( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  x.  1 )  =  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 ) )
5951, 58eqtrd 2484 . . . . . 6  |-  ( ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  /\  c  e.  (
<. V ,  E >. Neighbors  v
) )  ->  (
( # `  ( (
<. V ,  E >. Neighbors  v
)  \  { c } ) )  x.  1 )  =  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) )
6059sumeq2dv 13506 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  sum_ c  e.  (
<. V ,  E >. Neighbors  v
) ( ( # `  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) )  x.  1 )  = 
sum_ c  e.  (
<. V ,  E >. Neighbors  v
) ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 ) )
6153nn0cnd 10861 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  e.  CC )
6243a1i 11 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  1  e.  CC )
6361, 62subcld 9936 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( ( # `
 ( <. V ,  E >. Neighbors  v ) )  - 
1 )  e.  CC )
6463adantr 465 . . . . . 6  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 )  e.  CC )
65 fsumconst 13586 . . . . . 6  |-  ( ( ( <. V ,  E >. Neighbors 
v )  e.  Fin  /\  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  e.  CC )  ->  sum_ c  e.  (
<. V ,  E >. Neighbors  v
) ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 )  =  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  x.  (
( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) ) )
6648, 64, 65syl2anc 661 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  sum_ c  e.  (
<. V ,  E >. Neighbors  v
) ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 )  =  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  x.  (
( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) ) )
67 hashnbgravdg 24889 . . . . . . . 8  |-  ( ( V USGrph  E  /\  v  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E
) `  v )
)
68 eqeq1 2447 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 v )  =  ( # `  ( <. V ,  E >. Neighbors  v
) )  ->  (
( ( V VDeg  E
) `  v )  =  K  <->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  K ) )
6968eqcoms 2455 . . . . . . . . 9  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E ) `  v )  ->  (
( ( V VDeg  E
) `  v )  =  K  <->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  K ) )
70 id 22 . . . . . . . . . 10  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  K  -> 
( # `  ( <. V ,  E >. Neighbors  v
) )  =  K )
71 oveq1 6288 . . . . . . . . . 10  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  K  -> 
( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  =  ( K  -  1 ) )
7270, 71oveq12d 6299 . . . . . . . . 9  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  K  -> 
( ( # `  ( <. V ,  E >. Neighbors  v
) )  x.  (
( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) )  =  ( K  x.  ( K  -  1 ) ) )
7369, 72syl6bi 228 . . . . . . . 8  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E ) `  v )  ->  (
( ( V VDeg  E
) `  v )  =  K  ->  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  x.  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  -  1 ) )  =  ( K  x.  ( K  - 
1 ) ) ) )
7467, 73syl 16 . . . . . . 7  |-  ( ( V USGrph  E  /\  v  e.  V )  ->  (
( ( V VDeg  E
) `  v )  =  K  ->  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  x.  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  -  1 ) )  =  ( K  x.  ( K  - 
1 ) ) ) )
7574adantlr 714 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( (
( V VDeg  E ) `  v )  =  K  ->  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  x.  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) )  =  ( K  x.  ( K  -  1 ) ) ) )
7675imp 429 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  x.  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) )  =  ( K  x.  ( K  -  1 ) ) )
7760, 66, 763eqtrd 2488 . . . 4  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  sum_ c  e.  (
<. V ,  E >. Neighbors  v
) ( ( # `  ( ( <. V ,  E >. Neighbors  v )  \  {
c } ) )  x.  1 )  =  ( K  x.  ( K  -  1 ) ) )
7829, 47, 773eqtrd 2488 . . 3  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( # `  ( M `  v )
)  =  ( K  x.  ( K  - 
1 ) ) )
7978ex 434 . 2  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( M `  v )
)  =  ( K  x.  ( K  - 
1 ) ) ) )
8079ralrimiva 2857 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( M `  v )
)  =  ( K  x.  ( K  - 
1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   {crab 2797   _Vcvv 3095    \ cdif 3458    C_ wss 3461   {csn 4014   <.cop 4020   <.cotp 4022   U_ciun 4315  Disj wdisj 4407   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   Fincfn 7518   CCcc 9493   RRcr 9494   1c1 9496    x. cmul 9500    - cmin 9810   NN0cn0 10802   #chash 12386   sum_csu 13489   USGrph cusg 24306   Neighbors cnbgra 24393   2SPathOnOt c2spthot 24832   VDeg cvdg 24869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11092  df-rp 11231  df-xadd 11329  df-fz 11683  df-fzo 11806  df-seq 12089  df-exp 12148  df-hash 12387  df-word 12523  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-clim 13292  df-sum 13490  df-usgra 24309  df-nbgra 24396  df-wlk 24484  df-trail 24485  df-pth 24486  df-spth 24487  df-wlkon 24490  df-spthon 24493  df-2wlkonot 24834  df-2spthonot 24836  df-2spthsot 24837  df-vdgr 24870
This theorem is referenced by:  usgreghash2spot  25045
  Copyright terms: Public domain W3C validator