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Theorem usgreghash2spot 25668
Description: In a finite k-regular graph with N vertices there are N times " k choose 2 " paths with length 2, according to statement 8 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 2492 . . . . . . . . . 10  |-  ( s  =  t  ->  (
s  e.  ( V 2SPathOnOt  E )  <->  t  e.  ( V 2SPathOnOt  E ) ) )
2 fveq2 5872 . . . . . . . . . . . 12  |-  ( s  =  t  ->  ( 1st `  s )  =  ( 1st `  t
) )
32fveq2d 5876 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  t ) ) )
43eqeq1d 2422 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  a ) )
51, 4anbi12d 715 . . . . . . . . 9  |-  ( s  =  t  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) ) )
65cbvrabv 3077 . . . . . . . 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 4500 . . . . . . 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 25666 . . . . . 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 1025 . . . . 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 430 . . . 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 5876 . . 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 458 . . . . 5  |-  ( ( V  e.  Fin  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K )  ->  V  e.  Fin )
13 simpr 462 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  y  e.  V )
14 3xpfi 7840 . . . . . . . . 9  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  Fin )
15 rabexg 4566 . . . . . . . . 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 17 . . . . . . . 8  |-  ( V  e.  Fin  ->  { s  e.  ( ( V  X.  V )  X.  V )  |  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  y ) }  e.  _V )
1716ad2antrr 730 . . . . . . 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 2435 . . . . . . . . . 10  |-  ( a  =  y  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  s
) )  =  y ) )
1918anbi2d 708 . . . . . . . . 9  |-  ( a  =  y  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  y ) ) )
2019rabbidv 3070 . . . . . . . 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 2420 . . . . . . . 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 5953 . . . . . . 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 665 . . . . . 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 730 . . . . . . 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 7793 . . . . . . 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 17 . . . . . 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 2508 . . . . 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 3087 . . . . . . 7  |-  ( V  e.  Fin  ->  V  e.  _V )
2972spotmdisj 25667 . . . . . . 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 17 . . . . . 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 466 . . . . 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 13849 . . . 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 1168 . . 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 25665 . . . . . . . . 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 2812 . . . . . . . . 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 17 . . . . . . . 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 1025 . . . . . . 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 430 . . . . . 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 5872 . . . . . . . . 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 5876 . . . . . . . 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 2422 . . . . . . 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 3178 . . . . . 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 473 . . . . 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 13736 . . . 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 1009 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  V  e.  Fin )
46 usgfidegfi 25509 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v
)  e.  NN0 )
47 r19.26 2953 . . . . . . . . . . 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 2492 . . . . . . . . . . . . . 14  |-  ( ( ( V VDeg  E ) `
 v )  =  K  ->  ( (
( V VDeg  E ) `  v )  e.  NN0  <->  K  e.  NN0 ) )
4948biimpac 488 . . . . . . . . . . . . 13  |-  ( ( ( ( V VDeg  E
) `  v )  e.  NN0  /\  ( ( V VDeg  E ) `  v )  =  K )  ->  K  e.  NN0 )
5049ralimi 2816 . . . . . . . . . . . 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 3883 . . . . . . . . . . . . . 14  |-  ( ( V  =/=  (/)  /\  A. v  e.  V  K  e.  NN0 )  ->  E. v  e.  V  K  e.  NN0 )
52 nn0cn 10868 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
53 kcnktkm1cn 10039 . . . . . . . . . . . . . . . 16  |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5452, 53syl 17 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  CC )
5554rexlimivw 2912 . . . . . . . . . . . . . 14  |-  ( E. v  e.  V  K  e.  NN0  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5651, 55syl 17 . . . . . . . . . . . . 13  |-  ( ( V  =/=  (/)  /\  A. v  e.  V  K  e.  NN0 )  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5756expcom 436 . . . . . . . . . . . 12  |-  ( A. v  e.  V  K  e.  NN0  ->  ( V  =/=  (/)  ->  ( K  x.  ( K  -  1 ) )  e.  CC ) )
5850, 57syl 17 . . . . . . . . . . 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 216 . . . . . . . . . 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 435 . . . . . . . . 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 81 . . . . . . . 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 17 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( V  =/=  (/)  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( K  x.  ( K  -  1
) )  e.  CC ) ) )
6362ex 435 . . . . . 6  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( V  =/=  (/)  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K  -> 
( K  x.  ( K  -  1 ) )  e.  CC ) ) ) )
64633imp1 1218 . . . . 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 13818 . . . . 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 665 . . . 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 2461 . . 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 2465 . 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 435 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   E.wrex 2774   {crab 2777   _Vcvv 3078   (/)c0 3758   U_ciun 4293  Disj wdisj 4388   class class class wbr 4417    |-> cmpt 4475    X. cxp 4843   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   Fincfn 7568   CCcc 9526   1c1 9529    x. cmul 9533    - cmin 9849   NN0cn0 10858   #chash 12501   sum_csu 13719   USGrph cusg 24929   2SPathOnOt c2spthot 25455   VDeg cvdg 25492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-ot 4002  df-uni 4214  df-int 4250  df-iun 4295  df-disj 4389  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-oi 8016  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-xadd 11399  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-word 12640  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-sum 13720  df-usgra 24932  df-nbgra 25019  df-wlk 25107  df-trail 25108  df-pth 25109  df-spth 25110  df-wlkon 25113  df-spthon 25116  df-2wlkonot 25457  df-2spthonot 25459  df-2spthsot 25460  df-vdgr 25493
This theorem is referenced by:  frgregordn0  25669
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