MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgreghash2spot Structured version   Unicode version

Theorem usgreghash2spot 24774
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 2539 . . . . . . . . . 10  |-  ( s  =  t  ->  (
s  e.  ( V 2SPathOnOt  E )  <->  t  e.  ( V 2SPathOnOt  E ) ) )
2 fveq2 5866 . . . . . . . . . . . 12  |-  ( s  =  t  ->  ( 1st `  s )  =  ( 1st `  t
) )
32fveq2d 5870 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( 2nd `  ( 1st `  s
) )  =  ( 2nd `  ( 1st `  t ) ) )
43eqeq1d 2469 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  t
) )  =  a ) )
51, 4anbi12d 710 . . . . . . . . 9  |-  ( s  =  t  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) ) )
65cbvrabv 3112 . . . . . . . 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 4530 . . . . . . 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 24772 . . . . . 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 1016 . . . . 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 5870 . . 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 457 . . . . 5  |-  ( ( V  e.  Fin  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  K )  ->  V  e.  Fin )
13 simpr 461 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\ 
A. v  e.  V  ( ( V VDeg  E
) `  v )  =  K )  /\  y  e.  V )  ->  y  e.  V )
14 3xpfi 7792 . . . . . . . . 9  |-  ( V  e.  Fin  ->  (
( V  X.  V
)  X.  V )  e.  Fin )
15 rabexg 4597 . . . . . . . . 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 725 . . . . . . 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 2482 . . . . . . . . . 10  |-  ( a  =  y  ->  (
( 2nd `  ( 1st `  s ) )  =  a  <->  ( 2nd `  ( 1st `  s
) )  =  y ) )
1918anbi2d 703 . . . . . . . . 9  |-  ( a  =  y  ->  (
( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s
) )  =  a )  <->  ( s  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  s ) )  =  y ) ) )
2019rabbidv 3105 . . . . . . . 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 2467 . . . . . . . 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 5948 . . . . . . 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 661 . . . . . 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 725 . . . . . . 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 7744 . . . . . . 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 2555 . . . . 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 3122 . . . . . . 7  |-  ( V  e.  Fin  ->  V  e.  _V )
2972spotmdisj 24773 . . . . . . 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 465 . . . . 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 13599 . . . 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 1159 . . 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 24771 . . . . . . . . 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 2853 . . . . . . . . 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 1016 . . . . . . 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 5866 . . . . . . . . 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 5870 . . . . . . . 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 2469 . . . . . . 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 3213 . . . . . 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 471 . . . . 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 13488 . . . 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 1000 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  V  e.  Fin )
46 usgfidegfi 24614 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v
)  e.  NN0 )
47 r19.26 2989 . . . . . . . . . . 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 2539 . . . . . . . . . . . . . 14  |-  ( ( ( V VDeg  E ) `
 v )  =  K  ->  ( (
( V VDeg  E ) `  v )  e.  NN0  <->  K  e.  NN0 ) )
4948biimpac 486 . . . . . . . . . . . . 13  |-  ( ( ( ( V VDeg  E
) `  v )  e.  NN0  /\  ( ( V VDeg  E ) `  v )  =  K )  ->  K  e.  NN0 )
5049ralimi 2857 . . . . . . . . . . . 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 3917 . . . . . . . . . . . . . 14  |-  ( ( V  =/=  (/)  /\  A. v  e.  V  K  e.  NN0 )  ->  E. v  e.  V  K  e.  NN0 )
52 nn0cn 10805 . . . . . . . . . . . . . . . 16  |-  ( K  e.  NN0  ->  K  e.  CC )
53 kcnktkm1cn 9988 . . . . . . . . . . . . . . . 16  |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
5452, 53syl 16 . . . . . . . . . . . . . . 15  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  CC )
5554rexlimivw 2952 . . . . . . . . . . . . . 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 1209 . . . . 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 13568 . . . . 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 661 . . . 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 2508 . . 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 2512 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   (/)c0 3785   U_ciun 4325  Disj wdisj 4417   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   Fincfn 7516   CCcc 9490   1c1 9493    x. cmul 9497    - cmin 9805   NN0cn0 10795   #chash 12373   sum_csu 13471   USGrph cusg 24034   2SPathOnOt c2spthot 24560   VDeg cvdg 24597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-xadd 11319  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-word 12508  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-usgra 24037  df-nbgra 24124  df-wlk 24212  df-trail 24213  df-pth 24214  df-spth 24215  df-wlkon 24218  df-spthon 24221  df-2wlkonot 24562  df-2spthonot 24564  df-2spthsot 24565  df-vdgr 24598
This theorem is referenced by:  frgregordn0  24775
  Copyright terms: Public domain W3C validator