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Theorem usgreghash2spotv 30659
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 30658 . . . . . . . 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 1187 . . . . . . 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 5695 . . . . . 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 30275 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  v  e.  V )  ->  ( <. V ,  E >. Neighbors  v
)  e.  Fin )
653expa 1187 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( <. V ,  E >. Neighbors  v )  e.  Fin )
7 diffi 7543 . . . . . . . . . . 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 1187 . . . . . . . . 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 7390 . . . . . . . . . 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 2799 . . . . . . . 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 7599 . . . . . . . 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 23344 . . . . . . . . . . . 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 3492 . . . . . . . . . 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 4182 . . . . . . . . . 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 2799 . . . . . . . 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 30132 . . . . . . . . 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 4265 . . . . . . . 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 13285 . . . . . 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 2475 . . . . 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 23342 . . . . . . . . . 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 30131 . . . . . . . 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 13285 . . . . . 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 4557 . . . . . . . 8  |-  <. c ,  v ,  d
>.  e.  _V
40 hashsng 12136 . . . . . . . 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 13180 . . . . . 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 9340 . . . . . . 7  |-  1  e.  CC
44 fsumconst 13257 . . . . . . 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 2479 . . . . 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 13180 . . . 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 12169 . . . . . . . . 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 6106 . . . . . . 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 12126 . . . . . . . . . . 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 10637 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  e.  RR )
55 peano2rem 9675 . . . . . . . . 9  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  e.  RR  ->  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  e.  RR )
56 ax-1rid 9352 . . . . . . . . 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 2475 . . . . . 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 13180 . . . . 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 10638 . . . . . . . 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 9719 . . . . . . 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 13257 . . . . . 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 23581 . . . . . . . 8  |-  ( ( V USGrph  E  /\  v  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E
) `  v )
)
68 eqeq1 2449 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 v )  =  ( # `  ( <. V ,  E >. Neighbors  v
) )  ->  (
( ( V VDeg  E
) `  v )  =  K  <->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  K ) )
6968eqcoms 2446 . . . . . . . . 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 6098 . . . . . . . . . 10  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  K  -> 
( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  =  ( K  -  1 ) )
7270, 71oveq12d 6109 . . . . . . . . 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 2479 . . . 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 2479 . . 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 2799 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 965    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   _Vcvv 2972    \ cdif 3325    C_ wss 3328   {csn 3877   <.cop 3883   <.cotp 3885   U_ciun 4171  Disj wdisj 4262   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   Fincfn 7310   CCcc 9280   RRcr 9281   1c1 9283    x. cmul 9287    - cmin 9595   NN0cn0 10579   #chash 12103   sum_csu 13163   USGrph cusg 23264   Neighbors cnbgra 23329   VDeg cvdg 23563   2SPathOnOt c2spthot 30375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-disj 4263  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-xadd 11090  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-word 12229  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-usgra 23266  df-nbgra 23332  df-wlk 23415  df-trail 23416  df-pth 23417  df-spth 23418  df-wlkon 23421  df-spthon 23424  df-vdgr 23564  df-2wlkonot 30377  df-2spthonot 30379  df-2spthsot 30380
This theorem is referenced by:  usgreghash2spot  30662
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