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Theorem usgreghash2spot 28395
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 2495 . . . . . . . . . 10  |-  ( s  =  t  ->  (
s  e.  ( V 2SPathOnOt  E )  <->  t  e.  ( V 2SPathOnOt  E ) ) )
2 fveq2 5720 . . . . . . . . . . . 12  |-  ( s  =  t  ->  ( 1st `  s )  =  ( 1st `  t
) )
32fveq2d 5724 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  t ) ) )
43eqeq1d 2443 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  a ) )
51, 4anbi12d 692 . . . . . . . . 9  |-  ( s  =  t  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) ) )
65cbvrabv 2947 . . . . . . . 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 4284 . . . . . . 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 28393 . . . . . 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 977 . . . . 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 419 . . . 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 5724 . . 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 444 . . . . 5  |-  ( ( V  e.  Fin  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K )  ->  V  e.  Fin )
13 simpr 448 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  y  e.  V )
14 3xpfi 28072 . . . . . . . . 9  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  Fin )
15 rabexg 4345 . . . . . . . . 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 707 . . . . . . 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 2444 . . . . . . . . . 10  |-  ( a  =  y  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  s
) )  =  y ) )
1918anbi2d 685 . . . . . . . . 9  |-  ( a  =  y  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  y ) ) )
2019rabbidv 2940 . . . . . . . 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 2435 . . . . . . . 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 5796 . . . . . . 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 643 . . . . . 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 707 . . . . . . 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 7325 . . . . . . 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 2509 . . . . 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 2956 . . . . . . 7  |-  ( V  e.  Fin  ->  V  e.  _V )
2972spotmdisj 28394 . . . . . . 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 452 . . . . 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 12593 . . . 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 1120 . . 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 28392 . . . . . . . . 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 2769 . . . . . . . . 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 977 . . . . . . 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 419 . . . . . 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 5720 . . . . . . . . 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 5724 . . . . . . . 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 2443 . . . . . . 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 3043 . . . . . 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 458 . . . . 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 12489 . . . 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 961 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  V  e.  Fin )
46 usgfidegfi 28313 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v
)  e.  NN0 )
47 r19.26 2830 . . . . . . . . . . 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 2495 . . . . . . . . . . . . . 14  |-  ( ( ( V VDeg  E ) `
 v )  =  K  ->  ( (
( V VDeg  E ) `  v )  e.  NN0  <->  K  e.  NN0 ) )
4948biimpac 473 . . . . . . . . . . . . 13  |-  ( ( ( ( V VDeg  E
) `  v )  e.  NN0  /\  ( ( V VDeg  E ) `  v )  =  K )  ->  K  e.  NN0 )
5049ralimi 2773 . . . . . . . . . . . 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 3709 . . . . . . . . . . . . . 14  |-  ( ( V  =/=  (/)  /\  A. v  e.  V  K  e.  NN0 )  ->  E. v  e.  V  K  e.  NN0 )
52 nn0cn 10223 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
53 kcnktkm1cn 28074 . . . . . . . . . . . . . . . 16  |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5452, 53syl 16 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  CC )
5554rexlimivw 2818 . . . . . . . . . . . . . 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 425 . . . . . . . . . . . 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 205 . . . . . . . . . 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 424 . . . . . . . . 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 74 . . . . . . . 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 424 . . . . . 6  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( V  =/=  (/)  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K  -> 
( K  x.  ( K  -  1 ) )  e.  CC ) ) ) )
64633imp1 1166 . . . . 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 12565 . . . . 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 643 . . . 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 2467 . . 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 2471 . 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 424 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948   (/)c0 3620   U_ciun 4085  Disj wdisj 4174   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   Fincfn 7101   CCcc 8980   1c1 8983    x. cmul 8987    - cmin 9283   NN0cn0 10213   #chash 11610   sum_csu 12471   USGrph cusg 21357   VDeg cvdg 21656   2SPathOnOt c2spthot 28276
This theorem is referenced by:  frgregordn0  28396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-xadd 10703  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-word 11715  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-usgra 21359  df-nbgra 21425  df-wlk 21508  df-trail 21509  df-pth 21510  df-spth 21511  df-wlkon 21514  df-spthon 21517  df-vdgr 21657  df-2wlkonot 28278  df-2spthonot 28280  df-2spthsot 28281
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