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Theorem usgreghash2spotv 28529
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 28528 . . . . . . . 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 1154 . . . . . . 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 5735 . . . . . 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 28328 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  v  e.  V )  ->  ( <. V ,  E >. Neighbors  v
)  e.  Fin )
653expa 1154 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( <. V ,  E >. Neighbors  v )  e.  Fin )
7 diffi 7342 . . . . . . . . . . 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 1154 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( ( <. V ,  E >. Neighbors  v
)  \  { c } )  e.  Fin )
109adantr 453 . . . . . . . 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 7190 . . . . . . . . . 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 2791 . . . . . . . 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 7397 . . . . . . . 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 644 . . . . . . 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 21450 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  v )  C_  V
)
1716ad3antrrr 712 . . . . . . . . . . 11  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  c  e.  ( <. V ,  E >. Neighbors 
v ) )  -> 
( <. V ,  E >. Neighbors 
v )  C_  V
)
1817ssdifd 3485 . . . . . . . . . 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 4106 . . . . . . . . . 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 2791 . . . . . . . 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 449 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  v  e.  V )
23 otiunsndisj 28081 . . . . . . . . 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 4188 . . . . . . . 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 59 . . . . . . 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 12606 . . . . . 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 2470 . . . . 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 453 . . . 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 708 . . . . . . 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 21448 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( c  e.  ( <. V ,  E >. Neighbors 
v )  ->  c  e.  V ) )
3332ad3antrrr 712 . . . . . . . . 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 420 . . . . . . . 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 708 . . . . . . . 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 28080 . . . . . . . 8  |-  ( ( c  e.  V  /\  v  e.  V )  -> Disj  d  e.  ( ( <. V ,  E >. Neighbors  v
)  \  { c } ) { <. c ,  v ,  d
>. } )
3734, 35, 36syl2anc 644 . . . . . . 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 12606 . . . . . 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 4431 . . . . . . . 8  |-  <. c ,  v ,  d
>.  e.  _V
40 hashsng 11652 . . . . . . . 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 12502 . . . . . 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 9053 . . . . . . 7  |-  1  e.  CC
44 fsumconst 12578 . . . . . . 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 645 . . . . . 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 2474 . . . . 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 12502 . . . 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 453 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( <. V ,  E >. Neighbors  v )  e.  Fin )
49 hashdifsn 11684 . . . . . . . . 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 459 . . . . . . . 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 6099 . . . . . . 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 11644 . . . . . . . . . . 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 10280 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  e.  RR )
55 peano2rem 9372 . . . . . . . . 9  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  e.  RR  ->  ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  e.  RR )
56 ax-1rid 9065 . . . . . . . . 9  |-  ( ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  e.  RR  ->  ( ( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  x.  1 )  =  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 ) )
5754, 55, 563syl 19 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( (
( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  x.  1 )  =  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 ) )
5857ad2antrr 708 . . . . . . 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 2470 . . . . . 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 12502 . . . . 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 10281 . . . . . . . 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 9416 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( ( # `
 ( <. V ,  E >. Neighbors  v ) )  - 
1 )  e.  CC )
6463adantr 453 . . . . . 6  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( ( # `  ( <. V ,  E >. Neighbors 
v ) )  - 
1 )  e.  CC )
65 fsumconst 12578 . . . . . 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 644 . . . . 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 21687 . . . . . . . 8  |-  ( ( V USGrph  E  /\  v  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E
) `  v )
)
68 eqeq1 2444 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 v )  =  ( # `  ( <. V ,  E >. Neighbors  v
) )  ->  (
( ( V VDeg  E
) `  v )  =  K  <->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  K ) )
6968eqcoms 2441 . . . . . . . . 9  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E ) `  v )  ->  (
( ( V VDeg  E
) `  v )  =  K  <->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  K ) )
70 id 21 . . . . . . . . . 10  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  K  -> 
( # `  ( <. V ,  E >. Neighbors  v
) )  =  K )
71 oveq1 6091 . . . . . . . . . 10  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  K  -> 
( ( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 )  =  ( K  -  1 ) )
7270, 71oveq12d 6102 . . . . . . . . 9  |-  ( (
# `  ( <. V ,  E >. Neighbors  v ) )  =  K  -> 
( ( # `  ( <. V ,  E >. Neighbors  v
) )  x.  (
( # `  ( <. V ,  E >. Neighbors  v
) )  -  1 ) )  =  ( K  x.  ( K  -  1 ) ) )
7369, 72syl6bi 221 . . . . . . . 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 697 . . . . . 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 420 . . . . 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 2474 . . . 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 2474 . . 3  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( # `  ( M `  v )
)  =  ( K  x.  ( K  - 
1 ) ) )
7978ex 425 . 2  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( M `  v )
)  =  ( K  x.  ( K  - 
1 ) ) ) )
8079ralrimiva 2791 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958    \ cdif 3319    C_ wss 3322   {csn 3816   <.cop 3819   <.cotp 3820   U_ciun 4095  Disj wdisj 4185   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   Fincfn 7112   CCcc 8993   RRcr 8994   1c1 8996    x. cmul 9000    - cmin 9296   NN0cn0 10226   #chash 11623   sum_csu 12484   USGrph cusg 21370   Neighbors cnbgra 21435   VDeg cvdg 21669   2SPathOnOt c2spthot 28388
This theorem is referenced by:  usgreghash2spot  28532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4186  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-xadd 10716  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-word 11728  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-usgra 21372  df-nbgra 21438  df-wlk 21521  df-trail 21522  df-pth 21523  df-spth 21524  df-wlkon 21527  df-spthon 21530  df-vdgr 21670  df-2wlkonot 28390  df-2spthonot 28392  df-2spthsot 28393
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