Step |
Hyp |
Ref |
Expression |
1 |
|
erclwwlkn.w |
⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 ) |
2 |
|
erclwwlkn.r |
⊢ ∼ = { 〈 𝑡 , 𝑢 〉 ∣ ( 𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) } |
3 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
4 |
3
|
fusgrvtxfi |
⊢ ( 𝐺 ∈ FinUSGraph → ( Vtx ‘ 𝐺 ) ∈ Fin ) |
5 |
4
|
adantr |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → ( Vtx ‘ 𝐺 ) ∈ Fin ) |
6 |
1 2
|
hashclwwlkn0 |
⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( ♯ ‘ 𝑊 ) = Σ 𝑥 ∈ ( 𝑊 / ∼ ) ( ♯ ‘ 𝑥 ) ) |
7 |
5 6
|
syl |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → ( ♯ ‘ 𝑊 ) = Σ 𝑥 ∈ ( 𝑊 / ∼ ) ( ♯ ‘ 𝑥 ) ) |
8 |
|
fusgrusgr |
⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph ) |
9 |
|
usgrumgr |
⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UMGraph ) |
10 |
8 9
|
syl |
⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ UMGraph ) |
11 |
1 2
|
umgrhashecclwwlk |
⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ ) → ( 𝑥 ∈ ( 𝑊 / ∼ ) → ( ♯ ‘ 𝑥 ) = 𝑁 ) ) |
12 |
10 11
|
sylan |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → ( 𝑥 ∈ ( 𝑊 / ∼ ) → ( ♯ ‘ 𝑥 ) = 𝑁 ) ) |
13 |
12
|
imp |
⊢ ( ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) ∧ 𝑥 ∈ ( 𝑊 / ∼ ) ) → ( ♯ ‘ 𝑥 ) = 𝑁 ) |
14 |
13
|
sumeq2dv |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → Σ 𝑥 ∈ ( 𝑊 / ∼ ) ( ♯ ‘ 𝑥 ) = Σ 𝑥 ∈ ( 𝑊 / ∼ ) 𝑁 ) |
15 |
1 2
|
qerclwwlknfi |
⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑊 / ∼ ) ∈ Fin ) |
16 |
5 15
|
syl |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → ( 𝑊 / ∼ ) ∈ Fin ) |
17 |
|
prmnn |
⊢ ( 𝑁 ∈ ℙ → 𝑁 ∈ ℕ ) |
18 |
17
|
nncnd |
⊢ ( 𝑁 ∈ ℙ → 𝑁 ∈ ℂ ) |
19 |
18
|
adantl |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → 𝑁 ∈ ℂ ) |
20 |
|
fsumconst |
⊢ ( ( ( 𝑊 / ∼ ) ∈ Fin ∧ 𝑁 ∈ ℂ ) → Σ 𝑥 ∈ ( 𝑊 / ∼ ) 𝑁 = ( ( ♯ ‘ ( 𝑊 / ∼ ) ) · 𝑁 ) ) |
21 |
16 19 20
|
syl2anc |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → Σ 𝑥 ∈ ( 𝑊 / ∼ ) 𝑁 = ( ( ♯ ‘ ( 𝑊 / ∼ ) ) · 𝑁 ) ) |
22 |
7 14 21
|
3eqtrd |
⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ ) → ( ♯ ‘ 𝑊 ) = ( ( ♯ ‘ ( 𝑊 / ∼ ) ) · 𝑁 ) ) |