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Theorem rusgrnumwwlks 30335

Description: Induction step for rusgrnumwwlk 30336. (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 1214	. . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑃 ∈ 𝑉)
2		simpr3 1215	. . 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 30331	. . . 4 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
6	5	eqeq1d 2765	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) ↔ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)))
7	1, 2, 6	syl2anc 595	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) ↔ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)))
8		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	8	wwlksnredwwlkn0 30254	. . . . . . . . . . . 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 417	. . . . . . . . . . 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 1153	. . . . . . . . . 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 486	. . . . . . . . 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 411	. . . . . . . 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 3422	. . . . . . 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 485	. . . . . 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 6885	. . . . 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 1155	. . . . . . . . . . . . 13 ⊢ (((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) → (𝑦‘0) = 𝑃)
18	17	pm4.71ri 569	. . . . . . . . . . . 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 3187	. . . . . . . . . 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 6880	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥‘0) = (𝑦‘0))
22	21	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
23	22	rexrab 3659	. . . . . . . . . 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 292	. . . . . . . . 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 3422	. . . . . . . 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 485	. . . . . . 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 6885	. . . . . 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 1234	. . . . . . 7 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → 𝑉 ∈ Fin)
29	3	eleq1i 2854	. . . . . . . 8 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
30	29	biimpi 219	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin)
31		eqid 2763	. . . . . . . 8 ⊢ ((𝑁 + 1) WWalksN 𝐺) = ((𝑁 + 1) WWalksN 𝐺)
32		eqid 2763	. . . . . . . 8 ⊢ {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃}
33	31, 8, 32	hashwwlksnext 30272	. . . . . . 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 19	. . . . . 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 6880	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
36	35	eqeq1d 2765	. . . . . . . . 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 15753	. . . . . . 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 2802	. . . . 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 30330	. . . . . . . . . . 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 2769	. . . . . . . . . 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 6885	. . . . . . . . 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 486	. . . . . . . 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 3646	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → 𝑦 ∈ (𝑁 WWalksN 𝐺))
46	45	adantl 486	. . . . . . . . 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 30261	. . . . . . . . 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 18	. . . . . . . 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 29944	. . . . . . . . . 10 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑝 ∈ 𝑉 (♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
50		fveq1 6880	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
51	50	eqeq1d 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
52	51	elrab 3650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃))
53	3, 8	wwlknp 30201	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
54	53	adantr 485	. . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑦 ∈ Word 𝑉)
56		nn0p1gt0 12537	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ₀ → 0 < (𝑁 + 1))
57	56	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 0 < (𝑁 + 1))
58	57	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 0 < (𝑁 + 1))
59		breq2 5113	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑦) = (𝑁 + 1) → (0 < (♯‘𝑦) ↔ 0 < (𝑁 + 1)))
60	59	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (0 < (♯‘𝑦) ↔ 0 < (𝑁 + 1)))
61	58, 60	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 0 < (♯‘𝑦))
62		hashle00 14441	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → ((♯‘𝑦) ≤ 0 ↔ 𝑦 = ∅))
63		lencl 14575	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ Word 𝑉 → (♯‘𝑦) ∈ ℕ₀)
64	63	nn0red 12570	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ Word 𝑉 → (♯‘𝑦) ∈ ℝ)
65		0re 11214	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
66		lenlt 11292	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((♯‘𝑦) ≤ 0 ↔ ¬ 0 < (♯‘𝑦)))
67	66	bicomd 226	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (¬ 0 < (♯‘𝑦) ↔ (♯‘𝑦) ≤ 0))
68	64, 65, 67	sylancl 597	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 0 < (♯‘𝑦) ↔ (♯‘𝑦) ≤ 0))
69		nne 2962	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅))
71	62, 68, 70	3bitr4rd 315	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (♯‘𝑦)))
72	71	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (♯‘𝑦)))
73	72	con4bid 320	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ≠ ∅ ↔ 0 < (♯‘𝑦)))
74	61, 73	mpbird 260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑦 ≠ ∅)
75	55, 74	jca 520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅))
76	75	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
77	76	3adant3 1150	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
79	52, 78	sylbi 220	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
80	79	imp 411	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅))
81		lswcl 14610	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅) → (lastS‘𝑦) ∈ 𝑉)
82	80, 81	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (lastS‘𝑦) ∈ 𝑉)
83	82	ad2antrr 738	. . . . . . . . . . . . . 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 4699	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (lastS‘𝑦) → {𝑝, 𝑛} = {(lastS‘𝑦), 𝑛})
85	84	eleq1d 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (lastS‘𝑦) → ({𝑝, 𝑛} ∈ (Edg‘𝐺) ↔ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)))
86	85	rabbidv 3423	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (lastS‘𝑦) → {𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)} = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)})
87	86	fveqeq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = (lastS‘𝑦) → ((♯‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
88	87	rspcva 3579	. . . . . . . . . . . . . 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 599	. . . . . . . . . . . . 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 439	. . . . . . . . . . . 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 97	. . . . . . . . . . 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 1153	. . . . . . . . . 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 18	. . . . . . . . 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 430	. . . . . . . 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 2802	. . . . . . 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 15758	. . . . . 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 7417	. . . . . . . 8 ⊢ ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾↑𝑁) · 𝐾))
98	97	adantl 486	. . . . . . 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 30264	. . . . . . . . . . . 12 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
100	29, 99	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑉 ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
101	100	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑁 WWalksN 𝐺) ∈ Fin)
102	101	ad2antlr 739	. . . . . . . . 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 9227	. . . . . . . . 9 ⊢ ((𝑁 WWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
104	102, 103	syl 18	. . . . . . . 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 29923	. . . . . . . . . . . . 13 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
106		simp1 1154	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 𝑉 ∈ Fin)
107	105, 106	anim12i 624	. . . . . . . . . . . 12 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
108	3	isfusgr 29677	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
109	107, 108	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐺 ∈ FinUSGraph)
110		simpl 487	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐺 RegUSGraph 𝐾)
111		ne0i 4294	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
112	111	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 𝑉 ≠ ∅)
113	112	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑉 ≠ ∅)
114	3	frusgrnn0 29930	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ₀)
115	109, 110, 113, 114	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐾 ∈ ℕ₀)
116	115	nn0cnd 12571	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐾 ∈ ℂ)
117	116	adantr 485	. . . . . . . 8 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → 𝐾 ∈ ℂ)
118		fsumconst 15846	. . . . . . . 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 595	. . . . . . 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 14188	. . . . . . . 8 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝐾↑(𝑁 + 1)) = ((𝐾↑𝑁) · 𝐾))
121	120	adantr 485	. . . . . . 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 2808	. . . . . 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 2798	. . . . 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 2802	. . . 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 12548	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
126	125	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℕ₀)
127	126	adantl 486	. . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) ∈ ℕ₀)
128	3, 4	rusgrnumwwlklem 30331	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑁 + 1) ∈ ℕ₀) → (𝑃𝐿(𝑁 + 1)) = (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
129	128	eqeq1d 2765	. . . . . 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 595	. . . . 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 485	. . . 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 260	. . 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 417	. 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 243	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 ∅c0 4286 {cpr 4591 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 Fincfn 8939 ℂcc 11102 ℝcr 11103 0cc0 11104 1c1 11105 + caddc 11107 · cmul 11109 < clt 11247 ≤ cle 11248 ℕ₀cn0 12508 ℕ₀^*cxnn0 12581 ..^cfzo 13687 ↑cexp 14102 ♯chash 14371 Word cword 14555 lastSclsw 14604 prefix cpfx 14713 Σcsu 15742 Vtxcvtx 29355 Edgcedg 29406 USGraphcusgr 29508 FinUSGraphcfusgr 29675 RegUSGraph crusgr 29915 WWalksN cwwlksn 30184

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-disj 5077 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-oi 9468 df-dju 9892 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-n0 12509 df-xnn0 12582 df-z 12596 df-uz 12867 df-rp 13021 df-xadd 13142 df-fz 13540 df-fzo 13688 df-seq 14043 df-exp 14103 df-hash 14372 df-word 14556 df-lsw 14605 df-concat 14613 df-s1 14639 df-substr 14684 df-pfx 14714 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-clim 15544 df-sum 15743 df-vtx 29357 df-iedg 29358 df-edg 29407 df-uhgr 29417 df-ushgr 29418 df-upgr 29441 df-umgr 29442 df-uspgr 29509 df-usgr 29510 df-fusgr 29676 df-nbgr 29692 df-vtxdg 29825 df-rgr 29916 df-rusgr 29917 df-wwlks 30188 df-wwlksn 30189

This theorem is used by: rusgrnumwwlk 30336

W3C validator