rusgrnumwwlks - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rusgrnumwwlks		Structured version Visualization version GIF version

Theorem rusgrnumwwlks 28339

Description: Induction step for rusgrnumwwlk 28340. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.) (Proof shortened by AV, 27-May-2022.)

**Hypotheses**
Ref	Expression
rusgrnumwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
rusgrnumwwlk.l	⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ₀ ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))

**Assertion**
Ref	Expression
rusgrnumwwlks	⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))

Distinct variable groups: 𝑛,𝐺,𝑣,𝑤 𝑛,𝑁,𝑣,𝑤 𝑃,𝑛,𝑣,𝑤 𝑛,𝑉,𝑣,𝑤 𝑤,𝐾 Allowed substitution hints: 𝐾(𝑣,𝑛) 𝐿(𝑤,𝑣,𝑛)

Proof of Theorem rusgrnumwwlks Dummy variables 𝑖 𝑝 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr2 1194	. . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑃 ∈ 𝑉)
2		simpr3 1195	. . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑁 ∈ ℕ₀)
3		rusgrnumwwlk.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
4		rusgrnumwwlk.l	. . . . 5 ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ₀ ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
5	3, 4	rusgrnumwwlklem 28335	. . . 4 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
6	5	eqeq1d 2740	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) ↔ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)))
7	1, 2, 6	syl2anc 584	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) ↔ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)))
8		eqid 2738	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	8	wwlksnredwwlkn0 28261	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))))
10	9	ex 413	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)))))
11	10	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)))))
12	11	adantl 482	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)))))
13	12	imp 407	. . . . . . . 8 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))))
14	13	rabbidva 3413	. . . . . . 7 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))})
15	14	adantr 481	. . . . . 6 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))})
16	15	fveq2d 6778	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
17		simp2 1136	. . . . . . . . . . . . 13 ⊢ (((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) → (𝑦‘0) = 𝑃)
18	17	pm4.71ri 561	. . . . . . . . . . . 12 ⊢ (((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) ↔ ((𝑦‘0) = 𝑃 ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))))
19	18	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) → (((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) ↔ ((𝑦‘0) = 𝑃 ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)))))
20	19	rexbidva 3225	. . . . . . . . . 10 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑦‘0) = 𝑃 ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)))))
21		fveq1 6773	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥‘0) = (𝑦‘0))
22	21	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
23	22	rexrab 3633	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑦‘0) = 𝑃 ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))))
24	20, 23	bitr4di 289	. . . . . . . . 9 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) ↔ ∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))))
25	24	rabbidva 3413	. . . . . . . 8 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))})
26	25	adantr 481	. . . . . . 7 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))})
27	26	fveq2d 6778	. . . . . 6 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
28		simplr1 1214	. . . . . . 7 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → 𝑉 ∈ Fin)
29	3	eleq1i 2829	. . . . . . . 8 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
30	29	biimpi 215	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin)
31		eqid 2738	. . . . . . . 8 ⊢ ((𝑁 + 1) WWalksN 𝐺) = ((𝑁 + 1) WWalksN 𝐺)
32		eqid 2738	. . . . . . . 8 ⊢ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃}
33	31, 8, 32	hashwwlksnext 28279	. . . . . . 7 ⊢ ((Vtx‘𝐺) ∈ Fin → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = Σ𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
34	28, 30, 33	3syl 18	. . . . . 6 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = Σ𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
35		fveq1 6773	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
36	35	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑥‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
37	36	cbvrabv 3426	. . . . . . . 8 ⊢ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
38	37	sumeq1i 15410	. . . . . . 7 ⊢ Σ𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))})
39	38	a1i 11	. . . . . 6 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → Σ𝑦 ∈ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
40	27, 34, 39	3eqtrd 2782	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
41		rusgrnumwwlkslem 28334	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))})
42	41	eqcomd 2744	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))})
43	42	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
44	43	adantl 482	. . . . . . . 8 ⊢ ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
45		elrabi 3618	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → 𝑦 ∈ (𝑁 WWalksN 𝐺))
46	45	adantl 482	. . . . . . . . 9 ⊢ ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) → 𝑦 ∈ (𝑁 WWalksN 𝐺))
47	3, 8	wwlksnexthasheq 28268	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑁 WWalksN 𝐺) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
48	46, 47	syl 17	. . . . . . . 8 ⊢ ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
49	3	rusgrpropadjvtx 27952	. . . . . . . . . 10 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑝 ∈ 𝑉 (♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
50		fveq1 6773	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
51	50	eqeq1d 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
52	51	elrab 3624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃))
53	3, 8	wwlknp 28208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
54	53	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃) → (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
55		simpll 764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑦 ∈ Word 𝑉)
56		nn0p1gt0 12262	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ₀ → 0 < (𝑁 + 1))
57	56	3ad2ant3 1134	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 0 < (𝑁 + 1))
58	57	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 0 < (𝑁 + 1))
59		breq2 5078	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑦) = (𝑁 + 1) → (0 < (♯‘𝑦) ↔ 0 < (𝑁 + 1)))
60	59	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (0 < (♯‘𝑦) ↔ 0 < (𝑁 + 1)))
61	58, 60	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 0 < (♯‘𝑦))
62		hashle00 14115	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → ((♯‘𝑦) ≤ 0 ↔ 𝑦 = ∅))
63		lencl 14236	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ Word 𝑉 → (♯‘𝑦) ∈ ℕ₀)
64	63	nn0red 12294	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ Word 𝑉 → (♯‘𝑦) ∈ ℝ)
65		0re 10977	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
66		lenlt 11053	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((♯‘𝑦) ≤ 0 ↔ ¬ 0 < (♯‘𝑦)))
67	66	bicomd 222	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (¬ 0 < (♯‘𝑦) ↔ (♯‘𝑦) ≤ 0))
68	64, 65, 67	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 0 < (♯‘𝑦) ↔ (♯‘𝑦) ≤ 0))
69		nne 2947	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅))
71	62, 68, 70	3bitr4rd 312	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (♯‘𝑦)))
72	71	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (♯‘𝑦)))
73	72	con4bid 317	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ≠ ∅ ↔ 0 < (♯‘𝑦)))
74	61, 73	mpbird 256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑦 ≠ ∅)
75	55, 74	jca 512	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅))
76	75	ex 413	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
77	76	3adant3 1131	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
78	54, 77	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
79	52, 78	sylbi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
80	79	imp 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅))
81		lswcl 14271	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅) → (lastS‘𝑦) ∈ 𝑉)
82	80, 81	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (lastS‘𝑦) ∈ 𝑉)
83	82	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) ∧ ∀𝑝 ∈ 𝑉 (♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → (lastS‘𝑦) ∈ 𝑉)
84		preq1 4669	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (lastS‘𝑦) → {𝑝, 𝑛} = {(lastS‘𝑦), 𝑛})
85	84	eleq1d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (lastS‘𝑦) → ({𝑝, 𝑛} ∈ (Edg‘𝐺) ↔ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)))
86	85	rabbidv 3414	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (lastS‘𝑦) → {𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)} = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)})
87	86	fveqeq2d 6782	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (lastS‘𝑦) → ((♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
88	87	rspcva 3559	. . . . . . . . . . . . . 14 ⊢ (((lastS‘𝑦) ∈ 𝑉 ∧ ∀𝑝 ∈ 𝑉 (♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
89	83, 88	sylancom 588	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) ∧ ∀𝑝 ∈ 𝑉 (♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
90	89	exp41 435	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁) → (∀𝑝 ∈ 𝑉 (♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 → (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))))
91	90	com14 96	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ 𝑉 (♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁) → (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))))
92	91	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑝 ∈ 𝑉 (♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁) → (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))))
93	49, 92	syl 17	. . . . . . . . 9 ⊢ (𝐺 RegUSGraph 𝐾 → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁) → (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))))
94	93	imp41 426	. . . . . . . 8 ⊢ ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) → (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
95	44, 48, 94	3eqtrd 2782	. . . . . . 7 ⊢ ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = 𝐾)
96	95	sumeq2dv 15415	. . . . . 6 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾)
97		oveq1 7282	. . . . . . . 8 ⊢ ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾↑𝑁) · 𝐾))
98	97	adantl 482	. . . . . . 7 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾↑𝑁) · 𝐾))
99		wwlksnfi 28271	. . . . . . . . . . . 12 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
100	29, 99	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑉 ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
101	100	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑁 WWalksN 𝐺) ∈ Fin)
102	101	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (𝑁 WWalksN 𝐺) ∈ Fin)
103		rabfi 9044	. . . . . . . . 9 ⊢ ((𝑁 WWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
104	102, 103	syl 17	. . . . . . . 8 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
105		rusgrusgr 27931	. . . . . . . . . . . . 13 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
106		simp1 1135	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 𝑉 ∈ Fin)
107	105, 106	anim12i 613	. . . . . . . . . . . 12 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
108	3	isfusgr 27685	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
109	107, 108	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐺 ∈ FinUSGraph)
110		simpl 483	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐺 RegUSGraph 𝐾)
111		ne0i 4268	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
112	111	3ad2ant2 1133	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 𝑉 ≠ ∅)
113	112	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑉 ≠ ∅)
114	3	frusgrnn0 27938	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ₀)
115	109, 110, 113, 114	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐾 ∈ ℕ₀)
116	115	nn0cnd 12295	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐾 ∈ ℂ)
117	116	adantr 481	. . . . . . . 8 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → 𝐾 ∈ ℂ)
118		fsumconst 15502	. . . . . . . 8 ⊢ (({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin ∧ 𝐾 ∈ ℂ) → Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾 = ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾))
119	104, 117, 118	syl2anc 584	. . . . . . 7 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾 = ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾))
120	116, 2	expp1d 13865	. . . . . . . 8 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝐾↑(𝑁 + 1)) = ((𝐾↑𝑁) · 𝐾))
121	120	adantr 481	. . . . . . 7 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (𝐾↑(𝑁 + 1)) = ((𝐾↑𝑁) · 𝐾))
122	98, 119, 121	3eqtr4d 2788	. . . . . 6 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾 = (𝐾↑(𝑁 + 1)))
123	96, 122	eqtrd 2778	. . . . 5 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (𝐾↑(𝑁 + 1)))
124	16, 40, 123	3eqtrd 2782	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1)))
125		peano2nn0 12273	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
126	125	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℕ₀)
127	126	adantl 482	. . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) ∈ ℕ₀)
128	3, 4	rusgrnumwwlklem 28335	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑁 + 1) ∈ ℕ₀) → (𝑃𝐿(𝑁 + 1)) = (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
129	128	eqeq1d 2740	. . . . . 6 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑁 + 1) ∈ ℕ₀) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
130	1, 127, 129	syl2anc 584	. . . . 5 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
131	130	adantr 481	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
132	124, 131	mpbird 256	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)))
133	132	ex 413	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
134	7, 133	sylbid 239	1 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 ∅c0 4256 {cpr 4563 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 ℕ₀cn0 12233 ℕ₀^*cxnn0 12305 ..^cfzo 13382 ↑cexp 13782 ♯chash 14044 Word cword 14217 lastSclsw 14265 prefix cpfx 14383 Σcsu 15397 Vtxcvtx 27366 Edgcedg 27417 USGraphcusgr 27519 FinUSGraphcfusgr 27683 RegUSGraph crusgr 27923 WWalksN cwwlksn 28191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-rp 12731 df-xadd 12849 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-vtx 27368 df-iedg 27369 df-edg 27418 df-uhgr 27428 df-ushgr 27429 df-upgr 27452 df-umgr 27453 df-uspgr 27520 df-usgr 27521 df-fusgr 27684 df-nbgr 27700 df-vtxdg 27833 df-rgr 27924 df-rusgr 27925 df-wwlks 28195 df-wwlksn 28196

This theorem is referenced by: rusgrnumwwlk 28340

W3C validator