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Theorem numclwwlk3lem 25310
Description: Lemma for numclwwlk3 25311. (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 996 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  X  e.  V )  ->  X  e.  V )
2 eluzge2nn0 11121 . . . . 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 25283 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X F N )  =  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X } )
61, 2, 5syl2an 475 . . . 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 5852 . . 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 2655 . . . . . . . . . 10  |-  ( -.  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 )  <->  ( w `  ( N  -  2 ) )  =  ( w `  0 ) )
109anbi2i 692 . . . . . . . . 9  |-  ( ( ( w `  0
)  =  X  /\  -.  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  <-> 
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )
1110orbi2i 517 . . . . . . . 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 3098 . . . . 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 3766 . . . . 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 2513 . . . 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 5852 . . 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 25279 . . . . . . . 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 464 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N ) )
21 simpl2 998 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  V  e.  Fin )
22 usgrav 24540 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2322simprd 461 . . . . . . . . 9  |-  ( V USGrph  E  ->  E  e.  _V )
24233ad2ant1 1015 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  X  e.  V )  ->  E  e.  _V )
2524adantr 463 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  E  e.  _V )
262adantl 464 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  N  e.  NN0 )
27 clwwlknfi 24980 . . . . . . 7  |-  ( ( V  e.  Fin  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( V ClWWalksN  E ) `  N )  e.  Fin )
2821, 25, 26, 27syl3anc 1226 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( V ClWWalksN  E ) `  N
)  e.  Fin )
2920, 28eqeltrd 2542 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( C `  N )  e.  Fin )
30 rabfi 7737 . . . . 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 7737 . . . . 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 3767 . . . . 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 2651 . . . . . . . . . . . . 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 464 . . . . . . . . . . 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 411 . . . . . . . . 9  |-  ( -.  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
)  \/  -.  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) ) )
40 ianor 486 . . . . . . . . 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 2868 . . . . . 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 3806 . . . . . 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 2507 . . . 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 12433 . . . 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 1226 . . 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 2499 . 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 25303 . . . . 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 475 . . . 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 5852 . . 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 25289 . . . . 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 469 . . . 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 5852 . . 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 6288 . 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 2498 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    u. cun 3459    i^i cin 3460   (/)c0 3783   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Fincfn 7509   0cc0 9481    + caddc 9484    - cmin 9796   2c2 10581   NN0cn0 10791   ZZ>=cuz 11082   #chash 12387   lastS clsw 12519   USGrph cusg 24532   WWalksN cwwlkn 24880   ClWWalksN cclwwlkn 24951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-word 12526  df-usgra 24535  df-clwwlk 24953  df-clwwlkn 24954
This theorem is referenced by:  numclwwlk3  25311
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