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Theorem usgreghash2spot 28532
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 2498 . . . . . . . . . 10  |-  ( s  =  t  ->  (
s  e.  ( V 2SPathOnOt  E )  <->  t  e.  ( V 2SPathOnOt  E ) ) )
2 fveq2 5731 . . . . . . . . . . . 12  |-  ( s  =  t  ->  ( 1st `  s )  =  ( 1st `  t
) )
32fveq2d 5735 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  t ) ) )
43eqeq1d 2446 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  a ) )
51, 4anbi12d 693 . . . . . . . . 9  |-  ( s  =  t  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) ) )
65cbvrabv 2957 . . . . . . . 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 4295 . . . . . . 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 28530 . . . . . 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 978 . . . . 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 420 . . . 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 5735 . . 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 445 . . . . 5  |-  ( ( V  e.  Fin  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K )  ->  V  e.  Fin )
13 simpr 449 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  y  e.  V )
14 3xpfi 28110 . . . . . . . . 9  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  Fin )
15 rabexg 4356 . . . . . . . . 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 708 . . . . . . 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 2447 . . . . . . . . . 10  |-  ( a  =  y  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  s
) )  =  y ) )
1918anbi2d 686 . . . . . . . . 9  |-  ( a  =  y  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  y ) ) )
2019rabbidv 2950 . . . . . . . 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 2438 . . . . . . . 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 5807 . . . . . . 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 644 . . . . . 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 708 . . . . . . 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 7336 . . . . . . 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 2512 . . . . 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 2966 . . . . . . 7  |-  ( V  e.  Fin  ->  V  e.  _V )
2972spotmdisj 28531 . . . . . . 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 453 . . . . 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 12606 . . . 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 1121 . . 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 28529 . . . . . . . . 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 2779 . . . . . . . . 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 978 . . . . . . 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 420 . . . . . 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 5731 . . . . . . . . 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 5735 . . . . . . . 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 2446 . . . . . . 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 3053 . . . . . 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 459 . . . . 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 12502 . . . 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 962 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  V  e.  Fin )
46 usgfidegfi 28425 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v
)  e.  NN0 )
47 r19.26 2840 . . . . . . . . . . 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 2498 . . . . . . . . . . . . . 14  |-  ( ( ( V VDeg  E ) `
 v )  =  K  ->  ( (
( V VDeg  E ) `  v )  e.  NN0  <->  K  e.  NN0 ) )
4948biimpac 474 . . . . . . . . . . . . 13  |-  ( ( ( ( V VDeg  E
) `  v )  e.  NN0  /\  ( ( V VDeg  E ) `  v )  =  K )  ->  K  e.  NN0 )
5049ralimi 2783 . . . . . . . . . . . 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 3719 . . . . . . . . . . . . . 14  |-  ( ( V  =/=  (/)  /\  A. v  e.  V  K  e.  NN0 )  ->  E. v  e.  V  K  e.  NN0 )
52 nn0cn 10236 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
53 kcnktkm1cn 28112 . . . . . . . . . . . . . . . 16  |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5452, 53syl 16 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  CC )
5554rexlimivw 2828 . . . . . . . . . . . . . 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 426 . . . . . . . . . . . 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 206 . . . . . . . . . 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 425 . . . . . . . . 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 75 . . . . . . . 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 425 . . . . . 6  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( V  =/=  (/)  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K  -> 
( K  x.  ( K  -  1 ) )  e.  CC ) ) ) )
64633imp1 1167 . . . . 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 12578 . . . . 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 644 . . . 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 2470 . . 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 2474 . 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 425 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958   (/)c0 3630   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   1c1 8996    x. cmul 9000    - cmin 9296   NN0cn0 10226   #chash 11623   sum_csu 12484   USGrph cusg 21370   VDeg cvdg 21669   2SPathOnOt c2spthot 28388
This theorem is referenced by:  frgregordn0  28533
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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