rusgrnumwwlks - Mathbox for Alexander van der Vekens

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Theorem rusgrnumwwlks 41177

Description: Induction step for rusgrnumwwlk 41178. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)

**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 1061	. . 3 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑃 ∈ 𝑉)
2		simpr3 1062	. . 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 41173	. . . 4 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}))
6	5	eqeq1d 2612	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) ↔ (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)))
7	1, 2, 6	syl2anc 691	. 2 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) ↔ (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)))
8		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
9	8	wwlksnredwwlkn0 41102	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))))
10	9	ex 449	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)))))
11	10	3ad2ant3 1077	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)))))
12	11	adantl 481	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)))))
13	12	imp 444	. . . . . . . 8 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))))
14	13	rabbidva 3163	. . . . . . 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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))})
15	14	adantr 480	. . . . . 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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))})
16	15	fveq2d 6107	. . . . 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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}))
17		simp2 1055	. . . . . . . . . . . . 13 ⊢ (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)) → (𝑦‘0) = 𝑃)
18	17	pm4.71ri 663	. . . . . . . . . . . 12 ⊢ (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)) ↔ ((𝑦‘0) = 𝑃 ∧ ((𝑤 substr ⟨0, (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)) ↔ ((𝑦‘0) = 𝑃 ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)))))
20	19	rexbidva 3031	. . . . . . . . . 10 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)) ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑦‘0) = 𝑃 ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)))))
21		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥‘0) = (𝑦‘0))
22	21	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
23	22	rexrab 3337	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)) ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑦‘0) = 𝑃 ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))))
24	20, 23	syl6bbr 277	. . . . . . . . 9 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)) ↔ ∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))))
25	24	rabbidva 3163	. . . . . . . 8 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑤 substr ⟨0, (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))})
26	25	adantr 480	. . . . . . 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 substr ⟨0, (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))})
27	26	fveq2d 6107	. . . . . 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 substr ⟨0, (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}))
28		simplr1 1096	. . . . . . 7 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → 𝑉 ∈ Fin)
29	3	eleq1i 2679	. . . . . . . 8 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
30	29	biimpi 205	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin)
31		eqid 2610	. . . . . . . 8 ⊢ ((𝑁 + 1) WWalkSN 𝐺) = ((𝑁 + 1) WWalkSN 𝐺)
32		eqid 2610	. . . . . . . 8 ⊢ {𝑥 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑥‘0) = 𝑃}
33	31, 8, 32	hashwwlksnext 41120	. . . . . . 7 ⊢ ((Vtx‘𝐺) ∈ Fin → (#‘{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ∃𝑦 ∈ {𝑥 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (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 substr ⟨0, (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 substr ⟨0, (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}))
35		fveq1 6102	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
36	35	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑥‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
37	36	cbvrabv 3172	. . . . . . . 8 ⊢ {𝑥 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}
38	37	sumeq1i 14276	. . . . . . 7 ⊢ Σ𝑦 ∈ {𝑥 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑥‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (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 substr ⟨0, (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 substr ⟨0, (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}))
40	27, 34, 39	3eqtrd 2648	. . . . 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 substr ⟨0, (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}))
41		rusgranumwlklem0 26475	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))} = {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))})
42	41	eqcomd 2616	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))} = {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))})
43	42	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}) = (#‘{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}))
44	43	adantl 481	. . . . . . . 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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}) = (#‘{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}))
45		elrabi 3328	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} → 𝑦 ∈ (𝑁 WWalkSN 𝐺))
46	45	adantl 481	. . . . . . . . 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 41109	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑁 WWalkSN 𝐺) → (#‘{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}) = (#‘{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
49	3	rusgrpropadjvtx 40785	. . . . . . . . . 10 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ₀^* ∧ ∀𝑝 ∈ 𝑉 (#‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
50		fveq1 6102	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
51	50	eqeq1d 2612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
52	51	elrab 3331	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} ↔ (𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑦‘0) = 𝑃))
53	3, 8	wwlknp 41045	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑁 WWalkSN 𝐺) → (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
54	53	adantr 480	. . . . . . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑦 ∈ Word 𝑉)
56		nn0p1gt0 11199	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ₀ → 0 < (𝑁 + 1))
57	56	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 0 < (𝑁 + 1))
58	57	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 0 < (𝑁 + 1))
59		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑦) = (𝑁 + 1) → (0 < (#‘𝑦) ↔ 0 < (𝑁 + 1)))
60	59	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (0 < (#‘𝑦) ↔ 0 < (𝑁 + 1)))
61	58, 60	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 0 < (#‘𝑦))
62		hashle00 13049	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → ((#‘𝑦) ≤ 0 ↔ 𝑦 = ∅))
63		lencl 13179	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ Word 𝑉 → (#‘𝑦) ∈ ℕ₀)
64	63	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ Word 𝑉 → (#‘𝑦) ∈ ℝ)
65		0re 9919	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
66		lenlt 9995	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((#‘𝑦) ≤ 0 ↔ ¬ 0 < (#‘𝑦)))
67	66	bicomd 212	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (¬ 0 < (#‘𝑦) ↔ (#‘𝑦) ≤ 0))
68	64, 65, 67	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 0 < (#‘𝑦) ↔ (#‘𝑦) ≤ 0))
69		nne 2786	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
70	69	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅))
71	62, 68, 70	3bitr4rd 300	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (#‘𝑦)))
72	71	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (#‘𝑦)))
73	72	con4bid 306	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ≠ ∅ ↔ 0 < (#‘𝑦)))
74	61, 73	mpbird 246	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑦 ≠ ∅)
75	55, 74	jca 553	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅))
76	75	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
77	76	3adant3 1074	. . . . . . . . . . . . . . . . . . 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 206	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅)))
80	79	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑦 ∈ Word 𝑉 ∧ 𝑦 ≠ ∅))
81		lswcl 13208	. . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . 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 4212	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ( lastS ‘𝑦) → {𝑝, 𝑛} = {( lastS ‘𝑦), 𝑛})
85	84	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ( lastS ‘𝑦) → ({𝑝, 𝑛} ∈ (Edg‘𝐺) ↔ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)))
86	85	rabbidv 3164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ( lastS ‘𝑦) → {𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)} = {𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)})
87	86	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ( lastS ‘𝑦) → (#‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = (#‘{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
88	87	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ( lastS ‘𝑦) → ((#‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (#‘{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
89	88	rspcva 3280	. . . . . . . . . . . . . 14 ⊢ ((( lastS ‘𝑦) ∈ 𝑉 ∧ ∀𝑝 ∈ 𝑉 (#‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → (#‘{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
90	83, 89	sylancom 698	. . . . . . . . . . . . 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)
91	90	exp41 636	. . . . . . . . . . . 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))))
92	91	com14 94	. . . . . . . . . . 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))))
93	92	3ad2ant3 1077	. . . . . . . . . 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))))
94	49, 93	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))))
95	94	imp41 617	. . . . . . . 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)
96	44, 48, 95	3eqtrd 2648	. . . . . . 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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}) = 𝐾)
97	96	sumeq2dv 14281	. . . . . 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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}) = Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾)
98		oveq1 6556	. . . . . . . 8 ⊢ ((#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁) → ((#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾↑𝑁) · 𝐾))
99	98	adantl 481	. . . . . . 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))
100		wwlksnfi 41112	. . . . . . . . . . . 12 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalkSN 𝐺) ∈ Fin)
101	29, 100	sylbi 206	. . . . . . . . . . 11 ⊢ (𝑉 ∈ Fin → (𝑁 WWalkSN 𝐺) ∈ Fin)
102	101	3ad2ant1 1075	. . . . . . . . . 10 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑁 WWalkSN 𝐺) ∈ Fin)
103	102	ad2antlr 759	. . . . . . . . 9 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → (𝑁 WWalkSN 𝐺) ∈ Fin)
104		rabfi 8070	. . . . . . . . 9 ⊢ ((𝑁 WWalkSN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
105	103, 104	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)
106		rusgrusgr 40764	. . . . . . . . . . . . 13 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph )
107		simp1 1054	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 𝑉 ∈ Fin)
108	106, 107	anim12i 588	. . . . . . . . . . . 12 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
109	3	isfusgr 40537	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
110	108, 109	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐺 ∈ FinUSGraph )
111		simpl 472	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐺 RegUSGraph 𝐾)
112		ne0i 3880	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ 𝑉 → 𝑉 ≠ ∅)
113	112	3ad2ant2 1076	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → 𝑉 ≠ ∅)
114	113	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝑉 ≠ ∅)
115	3	frusgrnn0 40771	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ₀)
116	110, 111, 114, 115	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐾 ∈ ℕ₀)
117	116	nn0cnd 11230	. . . . . . . . 9 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → 𝐾 ∈ ℂ)
118	117	adantr 480	. . . . . . . 8 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) ∧ (#‘{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) → 𝐾 ∈ ℂ)
119		fsumconst 14364	. . . . . . . 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))
120	105, 118, 119	syl2anc 691	. . . . . . 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))
121	117, 2	expp1d 12871	. . . . . . . 8 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝐾↑(𝑁 + 1)) = ((𝐾↑𝑁) · 𝐾))
122	121	adantr 480	. . . . . . 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))
123	99, 120, 122	3eqtr4d 2654	. . . . . 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)))
124	97, 123	eqtrd 2644	. . . . 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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}) = (𝐾↑(𝑁 + 1)))
125	16, 40, 124	3eqtrd 2648	. . . 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)))
126		peano2nn0 11210	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
127	126	3ad2ant3 1077	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℕ₀)
128	127	adantl 481	. . . . . 6 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) ∈ ℕ₀)
129	3, 4	rusgrnumwwlklem 41173	. . . . . . 7 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑁 + 1) ∈ ℕ₀) → (𝑃𝐿(𝑁 + 1)) = (#‘{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}))
130	129	eqeq1d 2612	. . . . . 6 ⊢ ((𝑃 ∈ 𝑉 ∧ (𝑁 + 1) ∈ ℕ₀) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (#‘{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
131	1, 128, 130	syl2anc 691	. . . . 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))))
132	131	adantr 480	. . . 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))))
133	125, 132	mpbird 246	. . 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)))
134	133	ex 449	. 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))))
135	7, 134	sylbid 229	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ∅c0 3874 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 ℕ₀cn0 11169 ℕ₀^*cxnn0 11240 ..^cfzo 12334 ↑cexp 12722 #chash 12979 Word cword 13146 lastS clsw 13147 substr csubstr 13150 Σcsu 14264 Vtxcvtx 25673 Edgcedga 25792 USGraph cusgr 40379 FinUSGraph cfusgr 40535 RegUSGraph crusgr 40756 WWalkSN cwwlksn 41029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-vtx 25675 df-iedg 25676 df-uhgr 25724 df-ushgr 25725 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-fusgr 40536 df-nbgr 40554 df-vtxdg 40682 df-rgr 40757 df-rusgr 40758 df-wwlks 41033 df-wwlksn 41034

This theorem is referenced by: rusgrnumwwlk 41178

W3C validator