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Theorem numclwwlk3lem 24973
Description: Lemma for numclwwlk3 24974. (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 997 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  X  e.  V )  ->  X  e.  V )
2 eluzge2nn0 11124 . . . . 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 24946 . . . . 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 5856 . . 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 2642 . . . . . . . . . 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 3085 . . . . 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 3751 . . . . 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 2500 . . . 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 5856 . . 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 24942 . . . . . . . 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 999 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  V  e.  Fin )
22 usgrav 24203 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2322simprd 463 . . . . . . . . 9  |-  ( V USGrph  E  ->  E  e.  _V )
24233ad2ant1 1016 . . . . . . . 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 24643 . . . . . . 7  |-  ( ( V  e.  Fin  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( V ClWWalksN  E ) `  N )  e.  Fin )
2821, 25, 26, 27syl3anc 1227 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( V ClWWalksN  E ) `  N
)  e.  Fin )
2920, 28eqeltrd 2529 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( C `  N )  e.  Fin )
30 rabfi 7742 . . . . 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 7742 . . . . 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 3752 . . . . 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 2638 . . . . . . . . . . . . 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 2855 . . . . . 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 3789 . . . . . 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 2494 . . . 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 12424 . . . 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 1227 . . 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 2486 . 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 24966 . . . . 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 5856 . . 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 24952 . . . . 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 5856 . . 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 6295 . 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 2485 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   {crab 2795   _Vcvv 3093    u. cun 3456    i^i cin 3457   (/)c0 3767   class class class wbr 4433    |-> cmpt 4491   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   Fincfn 7514   0cc0 9490    + caddc 9493    - cmin 9805   2c2 10586   NN0cn0 10796   ZZ>=cuz 11085   #chash 12379   lastS clsw 12509   USGrph cusg 24195   WWalksN cwwlkn 24543   ClWWalksN cclwwlkn 24614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-word 12516  df-usgra 24198  df-clwwlk 24616  df-clwwlkn 24617
This theorem is referenced by:  numclwwlk3  24974
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