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Theorem numclwwlk3lem 25829
Description: Lemma for numclwwlk3 25830. (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 1009 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  X  e.  V )  ->  X  e.  V )
2 eluzge2nn0 11195 . . . . 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 25802 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  NN0 )  -> 
( X F N )  =  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  X } )
61, 2, 5syl2an 480 . . . 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 5867 . . 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 pm4.42 970 . . . . . . . 8  |-  ( ( w `  0 )  =  X  <->  ( (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  \/  ( ( w `
 0 )  =  X  /\  -.  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) ) )
9 nne 2627 . . . . . . . . . 10  |-  ( -.  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 )  <->  ( w `  ( N  -  2 ) )  =  ( w `  0 ) )
109anbi2i 699 . . . . . . . . 9  |-  ( ( ( w `  0
)  =  X  /\  -.  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  <-> 
( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =  ( w `  0 ) ) )
1110orbi2i 522 . . . . . . . 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 253 . . . . . . 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 3035 . . . . 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 3713 . . . . 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 2502 . . . 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 5867 . . 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 25798 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( C `
 N )  =  ( ( V ClWWalksN  E ) `
 N ) )
192, 18syl 17 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N ) )
2019adantl 468 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N ) )
21 simpl2 1011 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  V  e.  Fin )
22 usgrav 25058 . . . . . . . . . 10  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2322simprd 465 . . . . . . . . 9  |-  ( V USGrph  E  ->  E  e.  _V )
24233ad2ant1 1028 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  X  e.  V )  ->  E  e.  _V )
2524adantr 467 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  E  e.  _V )
262adantl 468 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  N  e.  NN0 )
27 clwwlknfi 25499 . . . . . . 7  |-  ( ( V  e.  Fin  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( V ClWWalksN  E ) `  N )  e.  Fin )
2821, 25, 26, 27syl3anc 1267 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( V ClWWalksN  E ) `  N
)  e.  Fin )
2920, 28eqeltrd 2528 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V )  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( C `  N )  e.  Fin )
30 rabfi 7793 . . . . 5  |-  ( ( C `  N )  e.  Fin  ->  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =/=  ( w ` 
0 ) ) }  e.  Fin )
3129, 30syl 17 . . . 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 7793 . . . . 5  |-  ( ( C `  N )  e.  Fin  ->  { w  e.  ( C `  N
)  |  ( ( w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) }  e.  Fin )
3329, 32syl 17 . . . 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 3714 . . . . 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 2623 . . . . . . . . . . . . 13  |-  ( ( w `  ( N  -  2 ) )  =/=  ( w ` 
0 )  <->  -.  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) )
3635biimpi 198 . . . . . . . . . . . 12  |-  ( ( w `  ( N  -  2 ) )  =/=  ( w ` 
0 )  ->  -.  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) )
3736adantl 468 . . . . . . . . . . 11  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  ->  -.  ( w `  ( N  -  2 ) )  =  ( w `  0 ) )
3837intnand 926 . . . . . . . . . 10  |-  ( ( ( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `
 0 ) )  ->  -.  ( (
w `  0 )  =  X  /\  (
w `  ( N  -  2 ) )  =  ( w ` 
0 ) ) )
3938imori 415 . . . . . . . . 9  |-  ( -.  ( ( w ` 
0 )  =  X  /\  ( w `  ( N  -  2
) )  =/=  (
w `  0 )
)  \/  -.  (
( w `  0
)  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `
 0 ) ) )
40 ianor 491 . . . . . . . . 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 213 . . . . . . . 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 2801 . . . . . 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 3753 . . . . . 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 216 . . . . 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 2496 . . . 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 12558 . . . 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 1267 . . 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 2488 . 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 25822 . . . . 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 480 . . . 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 5867 . . 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 25808 . . . . 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 474 . . . 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 5867 . . 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 6306 . 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 2487 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 188    \/ wo 370    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   {crab 2740   _Vcvv 3044    u. cun 3401    i^i cin 3402   (/)c0 3730   class class class wbr 4401    |-> cmpt 4460   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   Fincfn 7566   0cc0 9536    + caddc 9539    - cmin 9857   2c2 10656   NN0cn0 10866   ZZ>=cuz 11156   #chash 12512   lastS clsw 12654   USGrph cusg 25050   WWalksN cwwlkn 25399   ClWWalksN cclwwlkn 25470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-word 12661  df-usgra 25053  df-clwwlk 25472  df-clwwlkn 25473
This theorem is referenced by:  numclwwlk3  25830
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