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Theorem usgreghash2spot 30571
Description: In a finite k-regular graph with N vertices there are N times " k choose 2 " paths with length 2, according to the proof of the third claim in the proof of the friendship theorem 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 ordered triples, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
Assertion
Ref Expression
usgreghash2spot  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) ) )
Distinct variable groups:    v, E    v, K    v, V

Proof of Theorem usgreghash2spot
Dummy variables  a 
s  t  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2501 . . . . . . . . . 10  |-  ( s  =  t  ->  (
s  e.  ( V 2SPathOnOt  E )  <->  t  e.  ( V 2SPathOnOt  E ) ) )
2 fveq2 5688 . . . . . . . . . . . 12  |-  ( s  =  t  ->  ( 1st `  s )  =  ( 1st `  t
) )
32fveq2d 5692 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  t ) ) )
43eqeq1d 2449 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  a ) )
51, 4anbi12d 705 . . . . . . . . 9  |-  ( s  =  t  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) ) )
65cbvrabv 2969 . . . . . . . 8  |-  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) }  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t
) )  =  a ) }
76mpteq2i 4372 . . . . . . 7  |-  ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } )  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) } )
87usgreg2spot 30569 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K  ->  ( V 2SPathOnOt  E )  =  U_ y  e.  V  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y ) ) )
983adant3 1003 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( V 2SPathOnOt  E )  =  U_ y  e.  V  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) ) )
109imp 429 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( V 2SPathOnOt  E )  =  U_ y  e.  V  ( (
a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) )
1110fveq2d 5692 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( # `  U_ y  e.  V  ( (
a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) ) )
12 simpl 454 . . . . 5  |-  ( ( V  e.  Fin  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K )  ->  V  e.  Fin )
13 simpr 458 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  y  e.  V )
14 3xpfi 30075 . . . . . . . . 9  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  Fin )
15 rabexg 4439 . . . . . . . . 9  |-  ( ( ( V  X.  V
)  X.  V )  e.  Fin  ->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) }  e.  _V )
1614, 15syl 16 . . . . . . . 8  |-  ( V  e.  Fin  ->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) }  e.  _V )
1716ad2antrr 720 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) }  e.  _V )
18 eqeq2 2450 . . . . . . . . . 10  |-  ( a  =  y  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  s
) )  =  y ) )
1918anbi2d 698 . . . . . . . . 9  |-  ( a  =  y  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  y ) ) )
2019rabbidv 2962 . . . . . . . 8  |-  ( a  =  y  ->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) }  =  {
s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) } )
21 eqid 2441 . . . . . . . 8  |-  ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } )  =  ( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } )
2220, 21fvmptg 5769 . . . . . . 7  |-  ( ( y  e.  V  /\  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) }  e.  _V )  ->  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y )  =  {
s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) } )
2313, 17, 22syl2anc 656 . . . . . 6  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y )  =  { s  e.  ( ( V  X.  V )  X.  V
)  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  y ) } )
2414ad2antrr 720 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  (
( V  X.  V
)  X.  V )  e.  Fin )
25 rabfi 7533 . . . . . . 7  |-  ( ( ( V  X.  V
)  X.  V )  e.  Fin  ->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) }  e.  Fin )
2624, 25syl 16 . . . . . 6  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) }  e.  Fin )
2723, 26eqeltrd 2515 . . . . 5  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y )  e.  Fin )
28 elex 2979 . . . . . . 7  |-  ( V  e.  Fin  ->  V  e.  _V )
2972spotmdisj 30570 . . . . . . 7  |-  ( V  e.  _V  -> Disj  y  e.  V  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) )
3028, 29syl 16 . . . . . 6  |-  ( V  e.  Fin  -> Disj  y  e.  V  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) )
3130adantr 462 . . . . 5  |-  ( ( V  e.  Fin  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K )  -> Disj  y  e.  V  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) )
3212, 27, 31hashiun 13281 . . . 4  |-  ( ( V  e.  Fin  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K )  ->  ( # `
 U_ y  e.  V  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V
)  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y ) )  =  sum_ y  e.  V  ( # `  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y ) ) )
33323ad2antl2 1146 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  U_ y  e.  V  ( (
a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) )  = 
sum_ y  e.  V  ( # `  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) ) )
347usgreghash2spotv 30568 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( (
( V VDeg  E ) `  v )  =  K  ->  ( # `  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  v ) )  =  ( K  x.  ( K  - 
1 ) ) ) )
35 ralim 2785 . . . . . . . . 9  |-  ( A. v  e.  V  (
( ( V VDeg  E
) `  v )  =  K  ->  ( # `  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V
)  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  v ) )  =  ( K  x.  ( K  - 
1 ) ) )  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K  ->  A. v  e.  V  ( # `  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  v ) )  =  ( K  x.  ( K  -  1 ) ) ) )
3634, 35syl 16 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K  ->  A. v  e.  V  ( # `  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  v ) )  =  ( K  x.  ( K  - 
1 ) ) ) )
37363adant3 1003 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  A. v  e.  V  ( # `  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  v ) )  =  ( K  x.  ( K  -  1 ) ) ) )
3837imp 429 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  A. v  e.  V  ( # `  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  v ) )  =  ( K  x.  ( K  - 
1 ) ) )
39 fveq2 5688 . . . . . . . . 9  |-  ( v  =  y  ->  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  v )  =  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) )
4039fveq2d 5692 . . . . . . . 8  |-  ( v  =  y  ->  ( # `
 ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  v ) )  =  ( # `  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y ) ) )
4140eqeq1d 2449 . . . . . . 7  |-  ( v  =  y  ->  (
( # `  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  v ) )  =  ( K  x.  ( K  -  1 ) )  <->  ( # `  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y ) )  =  ( K  x.  ( K  - 
1 ) ) ) )
4241rspccva 3069 . . . . . 6  |-  ( ( A. v  e.  V  ( # `  ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  v ) )  =  ( K  x.  ( K  -  1 ) )  /\  y  e.  V )  ->  ( # `
 ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) )  =  ( K  x.  ( K  -  1 ) ) )
4338, 42sylan 468 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K )  /\  y  e.  V )  ->  ( # `
 ( ( a  e.  V  |->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a ) } ) `  y ) )  =  ( K  x.  ( K  -  1 ) ) )
4443sumeq2dv 13176 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  sum_ y  e.  V  ( # `  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y ) )  =  sum_ y  e.  V  ( K  x.  ( K  -  1 ) ) )
45 simpl2 987 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  V  e.  Fin )
46 usgfidegfi 30436 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v
)  e.  NN0 )
47 r19.26 2847 . . . . . . . . . . 11  |-  ( A. v  e.  V  (
( ( V VDeg  E
) `  v )  e.  NN0  /\  ( ( V VDeg  E ) `  v )  =  K )  <->  ( A. v  e.  V  ( ( V VDeg  E ) `  v
)  e.  NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K ) )
48 eleq1 2501 . . . . . . . . . . . . . 14  |-  ( ( ( V VDeg  E ) `
 v )  =  K  ->  ( (
( V VDeg  E ) `  v )  e.  NN0  <->  K  e.  NN0 ) )
4948biimpac 483 . . . . . . . . . . . . 13  |-  ( ( ( ( V VDeg  E
) `  v )  e.  NN0  /\  ( ( V VDeg  E ) `  v )  =  K )  ->  K  e.  NN0 )
5049ralimi 2789 . . . . . . . . . . . 12  |-  ( A. v  e.  V  (
( ( V VDeg  E
) `  v )  e.  NN0  /\  ( ( V VDeg  E ) `  v )  =  K )  ->  A. v  e.  V  K  e.  NN0 )
51 r19.2z 3766 . . . . . . . . . . . . . 14  |-  ( ( V  =/=  (/)  /\  A. v  e.  V  K  e.  NN0 )  ->  E. v  e.  V  K  e.  NN0 )
52 nn0cn 10585 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
53 kcnktkm1cn 30079 . . . . . . . . . . . . . . . 16  |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5452, 53syl 16 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  CC )
5554rexlimivw 2835 . . . . . . . . . . . . . 14  |-  ( E. v  e.  V  K  e.  NN0  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5651, 55syl 16 . . . . . . . . . . . . 13  |-  ( ( V  =/=  (/)  /\  A. v  e.  V  K  e.  NN0 )  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5756expcom 435 . . . . . . . . . . . 12  |-  ( A. v  e.  V  K  e.  NN0  ->  ( V  =/=  (/)  ->  ( K  x.  ( K  -  1 ) )  e.  CC ) )
5850, 57syl 16 . . . . . . . . . . 11  |-  ( A. v  e.  V  (
( ( V VDeg  E
) `  v )  e.  NN0  /\  ( ( V VDeg  E ) `  v )  =  K )  ->  ( V  =/=  (/)  ->  ( K  x.  ( K  -  1 ) )  e.  CC ) )
5947, 58sylbir 213 . . . . . . . . . 10  |-  ( ( A. v  e.  V  ( ( V VDeg  E
) `  v )  e.  NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( V  =/=  (/)  ->  ( K  x.  ( K  -  1
) )  e.  CC ) )
6059ex 434 . . . . . . . . 9  |-  ( A. v  e.  V  (
( V VDeg  E ) `  v )  e.  NN0  ->  ( A. v  e.  V  ( ( V VDeg 
E ) `  v
)  =  K  -> 
( V  =/=  (/)  ->  ( K  x.  ( K  -  1 ) )  e.  CC ) ) )
6160com23 78 . . . . . . . 8  |-  ( A. v  e.  V  (
( V VDeg  E ) `  v )  e.  NN0  ->  ( V  =/=  (/)  ->  ( A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K  ->  ( K  x.  ( K  - 
1 ) )  e.  CC ) ) )
6246, 61syl 16 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( V  =/=  (/)  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( K  x.  ( K  -  1
) )  e.  CC ) ) )
6362ex 434 . . . . . 6  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( V  =/=  (/)  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K  -> 
( K  x.  ( K  -  1 ) )  e.  CC ) ) ) )
64633imp1 1195 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
65 fsumconst 13253 . . . . 5  |-  ( ( V  e.  Fin  /\  ( K  x.  ( K  -  1 ) )  e.  CC )  ->  sum_ y  e.  V  ( K  x.  ( K  -  1 ) )  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) ) )
6645, 64, 65syl2anc 656 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  sum_ y  e.  V  ( K  x.  ( K  -  1
) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) )
6744, 66eqtrd 2473 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  sum_ y  e.  V  ( # `  (
( a  e.  V  |->  { s  e.  ( ( V  X.  V
)  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  a ) } ) `  y ) )  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) ) )
6811, 33, 673eqtrd 2477 . 2  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) )
6968ex 434 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970   (/)c0 3634   U_ciun 4168  Disj wdisj 4259   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   Fincfn 7306   CCcc 9276   1c1 9279    x. cmul 9283    - cmin 9591   NN0cn0 10575   #chash 12099   sum_csu 13159   USGrph cusg 23183   VDeg cvdg 23482   2SPathOnOt c2spthot 30284
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-xadd 11086  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-word 12225  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-usgra 23185  df-nbgra 23251  df-wlk 23334  df-trail 23335  df-pth 23336  df-spth 23337  df-wlkon 23340  df-spthon 23343  df-vdgr 23483  df-2wlkonot 30286  df-2spthonot 30288  df-2spthsot 30289
This theorem is referenced by:  frgregordn0  30572
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