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Theorem usgreghash2spotv 24729
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 24728 . . . . . . . 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 1191 . . . . . . 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 5861 . . . . . 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 24111 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  v  e.  V )  ->  ( <. V ,  E >. Neighbors  v
)  e.  Fin )
653expa 1191 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( <. V ,  E >. Neighbors  v )  e.  Fin )
7 diffi 7740 . . . . . . . . . . 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 1191 . . . . . . . . 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 7586 . . . . . . . . . 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 2871 . . . . . . . 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 7797 . . . . . . . 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 24095 . . . . . . . . . . . 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 3633 . . . . . . . . . 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 4330 . . . . . . . . . 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 2871 . . . . . . . 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 4746 . . . . . . . . 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 4413 . . . . . . . 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 13585 . . . . . 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 2501 . . . . 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 24093 . . . . . . . . . 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 4745 . . . . . . . 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 13585 . . . . . 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 4705 . . . . . . . 8  |-  <. c ,  v ,  d
>.  e.  _V
40 hashsng 12393 . . . . . . . 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 13474 . . . . . 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 9539 . . . . . . 7  |-  1  e.  CC
44 fsumconst 13554 . . . . . . 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 2505 . . . . 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 13474 . . . 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 12429 . . . . . . . . 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 6290 . . . . . . 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 12383 . . . . . . . . . . 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 10842 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  e.  RR )
55 peano2rem 9875 . . . . . . . . 9  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  e.  RR  ->  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  e.  RR )
56 ax-1rid 9551 . . . . . . . . 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 2501 . . . . . 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 13474 . . . . 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 10843 . . . . . . . 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 9919 . . . . . . 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 13554 . . . . . 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 24575 . . . . . . . 8  |-  ( ( V USGrph  E  /\  v  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E
) `  v )
)
68 eqeq1 2464 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 v )  =  ( # `  ( <. V ,  E >. Neighbors  v
) )  ->  (
( ( V VDeg  E
) `  v )  =  K  <->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  K ) )
6968eqcoms 2472 . . . . . . . . 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 6282 . . . . . . . . . 10  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  K  -> 
( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  =  ( K  -  1 ) )
7270, 71oveq12d 6293 . . . . . . . . 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 2505 . . . 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 2505 . . 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 2871 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 968    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   _Vcvv 3106    \ cdif 3466    C_ wss 3469   {csn 4020   <.cop 4026   <.cotp 4028   U_ciun 4318  Disj wdisj 4410   class class class wbr 4440    |-> cmpt 4498    X. cxp 4990   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   Fincfn 7506   CCcc 9479   RRcr 9480   1c1 9482    x. cmul 9486    - cmin 9794   NN0cn0 10784   #chash 12360   sum_csu 13457   USGrph cusg 23993   Neighbors cnbgra 24079   2SPathOnOt c2spthot 24518   VDeg cvdg 24555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-xadd 11308  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-word 12495  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-usgra 23996  df-nbgra 24082  df-wlk 24170  df-trail 24171  df-pth 24172  df-spth 24173  df-wlkon 24176  df-spthon 24179  df-2wlkonot 24520  df-2spthonot 24522  df-2spthsot 24523  df-vdgr 24556
This theorem is referenced by:  usgreghash2spot  24732
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