Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlkn.w |
|- W = ( N ClWWalksN G ) |
2 |
|
erclwwlkn.r |
|- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) } |
3 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
4 |
3
|
fusgrvtxfi |
|- ( G e. FinUSGraph -> ( Vtx ` G ) e. Fin ) |
5 |
4
|
adantr |
|- ( ( G e. FinUSGraph /\ N e. Prime ) -> ( Vtx ` G ) e. Fin ) |
6 |
1 2
|
hashclwwlkn0 |
|- ( ( Vtx ` G ) e. Fin -> ( # ` W ) = sum_ x e. ( W /. .~ ) ( # ` x ) ) |
7 |
5 6
|
syl |
|- ( ( G e. FinUSGraph /\ N e. Prime ) -> ( # ` W ) = sum_ x e. ( W /. .~ ) ( # ` x ) ) |
8 |
|
fusgrusgr |
|- ( G e. FinUSGraph -> G e. USGraph ) |
9 |
|
usgrumgr |
|- ( G e. USGraph -> G e. UMGraph ) |
10 |
8 9
|
syl |
|- ( G e. FinUSGraph -> G e. UMGraph ) |
11 |
1 2
|
umgrhashecclwwlk |
|- ( ( G e. UMGraph /\ N e. Prime ) -> ( x e. ( W /. .~ ) -> ( # ` x ) = N ) ) |
12 |
10 11
|
sylan |
|- ( ( G e. FinUSGraph /\ N e. Prime ) -> ( x e. ( W /. .~ ) -> ( # ` x ) = N ) ) |
13 |
12
|
imp |
|- ( ( ( G e. FinUSGraph /\ N e. Prime ) /\ x e. ( W /. .~ ) ) -> ( # ` x ) = N ) |
14 |
13
|
sumeq2dv |
|- ( ( G e. FinUSGraph /\ N e. Prime ) -> sum_ x e. ( W /. .~ ) ( # ` x ) = sum_ x e. ( W /. .~ ) N ) |
15 |
1 2
|
qerclwwlknfi |
|- ( ( Vtx ` G ) e. Fin -> ( W /. .~ ) e. Fin ) |
16 |
5 15
|
syl |
|- ( ( G e. FinUSGraph /\ N e. Prime ) -> ( W /. .~ ) e. Fin ) |
17 |
|
prmnn |
|- ( N e. Prime -> N e. NN ) |
18 |
17
|
nncnd |
|- ( N e. Prime -> N e. CC ) |
19 |
18
|
adantl |
|- ( ( G e. FinUSGraph /\ N e. Prime ) -> N e. CC ) |
20 |
|
fsumconst |
|- ( ( ( W /. .~ ) e. Fin /\ N e. CC ) -> sum_ x e. ( W /. .~ ) N = ( ( # ` ( W /. .~ ) ) x. N ) ) |
21 |
16 19 20
|
syl2anc |
|- ( ( G e. FinUSGraph /\ N e. Prime ) -> sum_ x e. ( W /. .~ ) N = ( ( # ` ( W /. .~ ) ) x. N ) ) |
22 |
7 14 21
|
3eqtrd |
|- ( ( G e. FinUSGraph /\ N e. Prime ) -> ( # ` W ) = ( ( # ` ( W /. .~ ) ) x. N ) ) |