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Theorem numclwwlk3lem 24900
Description: Lemma for numclwwlk3 24901. (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 998 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  X  e.  V )  ->  X  e.  V )
2 eluzge2nn0 11131 . . . . 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 24873 . . . . 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 5875 . . 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 pm4.42 958 . . . . . . . 8  |-  ( ( w `  0 )  =  X  <->  ( (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  \/  ( ( w `
 0 )  =  X  /\  -.  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) ) )
9 nne 2668 . . . . . . . . . 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 3110 . . . . 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 3774 . . . . 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 2526 . . . 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 5875 . . 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 24869 . . . . . . . 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 1000 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  V  e.  Fin )
22 usgrav 24129 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2322simprd 463 . . . . . . . . 9  |-  ( V USGrph  E  ->  E  e.  _V )
24233ad2ant1 1017 . . . . . . . 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 24569 . . . . . . 7  |-  ( ( V  e.  Fin  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( V ClWWalksN  E ) `  N )  e.  Fin )
2821, 25, 26, 27syl3anc 1228 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( V ClWWalksN  E ) `  N
)  e.  Fin )
2920, 28eqeltrd 2555 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( C `  N )  e.  Fin )
30 rabfi 7754 . . . . 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 7754 . . . . 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 3775 . . . . 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 2664 . . . . . . . . . . . . 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 914 . . . . . . . . . 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 2881 . . . . . 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 3812 . . . . . 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 2520 . . . 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 12428 . . . 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 1228 . . 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 2512 . 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 24893 . . . . 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 5875 . . 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 24879 . . . . 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 5875 . . 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 6312 . 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 2511 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   {crab 2821   _Vcvv 3118    u. cun 3479    i^i cin 3480   (/)c0 3790   class class class wbr 4452    |-> cmpt 4510   ` cfv 5593  (class class class)co 6294    |-> cmpt2 6296   Fincfn 7526   0cc0 9502    + caddc 9505    - cmin 9815   2c2 10595   NN0cn0 10805   ZZ>=cuz 11092   #chash 12383   lastS clsw 12511   USGrph cusg 24121   WWalksN cwwlkn 24469   ClWWalksN cclwwlkn 24540
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-card 8330  df-cda 8558  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-seq 12086  df-exp 12145  df-hash 12384  df-word 12518  df-usgra 24124  df-clwwlk 24542  df-clwwlkn 24543
This theorem is referenced by:  numclwwlk3  24901
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