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Theorem numclwwlk3lem 30706
Description: Lemma for numclwwlk3 30707. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
numclwwlk.g  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
numclwwlk.q  |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  (
( V WWalksN  E ) `  n )  |  ( ( w `  0
)  =  v  /\  ( lastS  `  w )  =/=  v ) } )
numclwwlk.h  |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) } )
Assertion
Ref Expression
numclwwlk3lem  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( # `  ( X F N ) )  =  ( ( # `  ( X H N ) )  +  (
# `  ( X G N ) ) ) )
Distinct variable groups:    n, E    n, N    n, V    w, C    w, N    C, n, v, w    v, N    n, X, v, w    v, V   
w, E    w, V    w, F    w, Q    w, G    v, E    v, H, w
Allowed substitution hints:    Q( v, n)    F( v, n)    G( v, n)    H( n)

Proof of Theorem numclwwlk3lem
StepHypRef Expression
1 simp3 990 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  X  e.  V )  ->  X  e.  V )
2 eluzge2nn0 30197 . . . . 5  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN0 )
3 numclwwlk.c . . . . . 6  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
4 numclwwlk.f . . . . . 6  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
53, 4numclwwlkovf 30679 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X F N )  =  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X } )
61, 2, 5syl2an 477 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( X F N )  =  {
w  e.  ( C `
 N )  |  ( w `  0
)  =  X }
)
76fveq2d 5700 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( # `  ( X F N ) )  =  ( # `  {
w  e.  ( C `
 N )  |  ( w `  0
)  =  X }
) )
8 pm4.42 951 . . . . . . . 8  |-  ( ( w `  0 )  =  X  <->  ( (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  \/  ( ( w `
 0 )  =  X  /\  -.  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) ) )
9 nne 2617 . . . . . . . . . 10  |-  ( -.  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 )  <->  ( w `  ( N  -  2 ) )  =  ( w `  0 ) )
109anbi2i 694 . . . . . . . . 9  |-  ( ( ( w `  0
)  =  X  /\  -.  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  <-> 
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )
1110orbi2i 519 . . . . . . . 8  |-  ( ( ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
)  \/  ( ( w `  0 )  =  X  /\  -.  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) ) )  <->  ( ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) )  \/  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) ) )
128, 11bitri 249 . . . . . . 7  |-  ( ( w `  0 )  =  X  <->  ( (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  \/  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) ) ) )
1312a1i 11 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( (
w `  0 )  =  X  <->  ( ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) )  \/  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) ) ) )
1413rabbidv 2969 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  =  {
w  e.  ( C `
 N )  |  ( ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) )  \/  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) ) ) } )
15 unrab 3626 . . . . 5  |-  ( { w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
) }  u.  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } )  =  { w  e.  ( C `  N )  |  ( ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) )  \/  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) ) }
1614, 15syl6eqr 2493 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X }  =  ( { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) ) }  u.  { w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } ) )
1716fveq2d 5700 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( # `  {
w  e.  ( C `
 N )  |  ( w `  0
)  =  X }
)  =  ( # `  ( { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) }  u.  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } ) ) )
183numclwwlkfvc 30675 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
192, 18syl 16 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N ) )
2019adantl 466 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N ) )
21 simpl2 992 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  V  e.  Fin )
22 usgrav 23275 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2322simprd 463 . . . . . . . . 9  |-  ( V USGrph  E  ->  E  e.  _V )
24233ad2ant1 1009 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  X  e.  V )  ->  E  e.  _V )
2524adantr 465 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  E  e.  _V )
262adantl 466 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  N  e.  NN0 )
27 clwwlknfi 30482 . . . . . . 7  |-  ( ( V  e.  Fin  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( V ClWWalksN  E ) `  N )  e.  Fin )
2821, 25, 26, 27syl3anc 1218 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( V ClWWalksN  E ) `  N
)  e.  Fin )
2920, 28eqeltrd 2517 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( C `  N )  e.  Fin )
30 rabfi 7542 . . . . 5  |-  ( ( C `  N )  e.  Fin  ->  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) }  e.  Fin )
3129, 30syl 16 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) }  e.  Fin )
32 rabfi 7542 . . . . 5  |-  ( ( C `  N )  e.  Fin  ->  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  e.  Fin )
3329, 32syl 16 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  e.  Fin )
34 inrab 3627 . . . . 5  |-  ( { w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
) }  i^i  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } )  =  { w  e.  ( C `  N )  |  ( ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) )  /\  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) ) }
35 df-ne 2613 . . . . . . . . . . . . 13  |-  ( ( w `  ( N  -  2 ) )  =/=  ( w ` 
0 )  <->  -.  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) )
3635biimpi 194 . . . . . . . . . . . 12  |-  ( ( w `  ( N  -  2 ) )  =/=  ( w ` 
0 )  ->  -.  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )
3736adantl 466 . . . . . . . . . . 11  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  ->  -.  ( w `  ( N  -  2 ) )  =  ( w `  0 ) )
3837intnand 907 . . . . . . . . . 10  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  ->  -.  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )
3938imori 413 . . . . . . . . 9  |-  ( -.  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
)  \/  -.  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) ) )
40 ianor 488 . . . . . . . . 9  |-  ( -.  ( ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) )  /\  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) ) )  <->  ( -.  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  \/  -.  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) ) )
4139, 40mpbir 209 . . . . . . . 8  |-  -.  (
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
)  /\  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )
4241a1i 11 . . . . . . 7  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  (
ZZ>= `  2 ) )  /\  w  e.  ( C `  N ) )  ->  -.  (
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
)  /\  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) ) )
4342ralrimiva 2804 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  A. w  e.  ( C `  N
)  -.  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  /\  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) ) ) )
44 rabeq0 3664 . . . . . 6  |-  ( { w  e.  ( C `
 N )  |  ( ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) )  /\  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) ) ) }  =  (/)  <->  A. w  e.  ( C `  N )  -.  (
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
)  /\  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) ) )
4543, 44sylibr 212 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  { w  e.  ( C `  N
)  |  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  /\  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) ) ) }  =  (/) )
4634, 45syl5eq 2487 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
) }  i^i  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } )  =  (/) )
47 hashun 12150 . . . 4  |-  ( ( { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) ) }  e.  Fin  /\  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  e.  Fin  /\  ( { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) ) }  i^i  { w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } )  =  (/) )  ->  ( # `  ( { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) }  u.  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } ) )  =  ( ( # `  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
) } )  +  ( # `  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } ) ) )
4831, 33, 46, 47syl3anc 1218 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( # `  ( { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) ) }  u.  { w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } ) )  =  ( ( # `  { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) ) } )  +  ( # `  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } ) ) )
497, 17, 483eqtrd 2479 . 2  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( # `  ( X F N ) )  =  ( ( # `  { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =/=  ( w `  0
) ) } )  +  ( # `  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } ) ) )
50 numclwwlk.g . . . . . 6  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
51 numclwwlk.q . . . . . 6  |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  (
( V WWalksN  E ) `  n )  |  ( ( w `  0
)  =  v  /\  ( lastS  `  w )  =/=  v ) } )
52 numclwwlk.h . . . . . 6  |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) } )
533, 4, 50, 51, 52numclwwlkovh 30699 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X H N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) } )
541, 2, 53syl2an 477 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( X H N )  =  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
) } )
5554fveq2d 5700 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( # `  ( X H N ) )  =  ( # `  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
) } ) )
563, 4, 50numclwwlkovg 30685 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( X G N )  =  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) } )
571, 56sylan 471 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( X G N )  =  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } )
5857fveq2d 5700 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( # `  ( X G N ) )  =  ( # `  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) } ) )
5955, 58oveq12d 6114 . 2  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( # `
 ( X H N ) )  +  ( # `  ( X G N ) ) )  =  ( (
# `  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) } )  +  ( # `  { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  X  /\  ( w `
 ( N  - 
2 ) )  =  ( w `  0
) ) } ) ) )
6049, 59eqtr4d 2478 1  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( # `  ( X F N ) )  =  ( ( # `  ( X H N ) )  +  (
# `  ( X G N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977    u. cun 3331    i^i cin 3332   (/)c0 3642   class class class wbr 4297    e. cmpt 4355   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   Fincfn 7315   0cc0 9287    + caddc 9290    - cmin 9600   2c2 10376   NN0cn0 10584   ZZ>=cuz 10866   #chash 12108   lastS clsw 12227   USGrph cusg 23269   WWalksN cwwlkn 30317   ClWWalksN cclwwlkn 30419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-word 12234  df-usgra 23271  df-clwwlk 30421  df-clwwlkn 30422
This theorem is referenced by:  numclwwlk3  30707
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