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Theorem rusgrnumwwlks 26941
Description: Induction step for rusgrnumwwlk 26942. (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 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
Assertion
Ref Expression
rusgrnumwwlks ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) → (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.
StepHypRef Expression
1 simpr2 1088 . . 3 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑃𝑉)
2 simpr3 1089 . . 3 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑁 ∈ ℕ0)
3 rusgrnumwwlk.v . . . . 5 𝑉 = (Vtx‘𝐺)
4 rusgrnumwwlk.l . . . . 5 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
53, 4rusgrnumwwlklem 26937 . . . 4 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
65eqeq1d 2653 . . 3 ((𝑃𝑉𝑁 ∈ ℕ0) → ((𝑃𝐿𝑁) = (𝐾𝑁) ↔ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)))
71, 2, 6syl2anc 694 . 2 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) ↔ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)))
8 eqid 2651 . . . . . . . . . . . . 13 (Edg‘𝐺) = (Edg‘𝐺)
98wwlksnredwwlkn0 26859 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑤 ∈ ((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))))
109ex 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)))))
11103ad2ant3 1104 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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)))))
1211adantl 481 . . . . . . . . 9 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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)))))
1312imp 444 . . . . . . . 8 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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))))
1413rabbidva 3219 . . . . . . 7 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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))})
1514adantr 480 . . . . . 6 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → {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))})
1615fveq2d 6233 . . . . 5 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{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 1082 . . . . . . . . . . . . 13 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺)) → (𝑦‘0) = 𝑃)
1817pm4.71ri 666 . . . . . . . . . . . 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))))
1918a1i 11 . . . . . . . . . . 11 ((((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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)))))
2019rexbidva 3078 . . . . . . . . . 10 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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 6228 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥‘0) = (𝑦‘0))
2221eqeq1d 2653 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
2322rexrab 3403 . . . . . . . . . 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))))
2420, 23syl6bbr 278 . . . . . . . . 9 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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))))
2524rabbidva 3219 . . . . . . . 8 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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))})
2625adantr 480 . . . . . . 7 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → {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))})
2726fveq2d 6233 . . . . . 6 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{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 1123 . . . . . . 7 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 𝑉 ∈ Fin)
293eleq1i 2721 . . . . . . . 8 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
3029biimpi 206 . . . . . . 7 (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin)
31 eqid 2651 . . . . . . . 8 ((𝑁 + 1) WWalksN 𝐺) = ((𝑁 + 1) WWalksN 𝐺)
32 eqid 2651 . . . . . . . 8 {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃}
3331, 8, 32hashwwlksnext 26877 . . . . . . 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))}))
3428, 30, 333syl 18 . . . . . 6 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{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 6228 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
3635eqeq1d 2653 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑥‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
3736cbvrabv 3230 . . . . . . . 8 {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
3837sumeq1i 14472 . . . . . . 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))})
3938a1i 11 . . . . . 6 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 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))}))
4027, 34, 393eqtrd 2689 . . . . 5 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{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 rusgrnumwwlkslem 26936 . . . . . . . . . . 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))})
4241eqcomd 2657 . . . . . . . . . 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))})
4342fveq2d 6233 . . . . . . . . 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))}))
4443adantl 481 . . . . . . . 8 ((((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 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 3391 . . . . . . . . . 10 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → 𝑦 ∈ (𝑁 WWalksN 𝐺))
4645adantl 481 . . . . . . . . 9 ((((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) → 𝑦 ∈ (𝑁 WWalksN 𝐺))
473, 8wwlksnexthasheq 26866 . . . . . . . . 9 (𝑦 ∈ (𝑁 WWalksN 𝐺) → (#‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ (Edg‘𝐺))}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
4846, 47syl 17 . . . . . . . 8 ((((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 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))}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
493rusgrpropadjvtx 26537 . . . . . . . . . 10 (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
50 fveq1 6228 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
5150eqeq1d 2653 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
5251elrab 3396 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃))
533, 8wwlknp 26791 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5453adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃) → (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
55 simpll 805 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑦 ∈ Word 𝑉)
56 nn0p1gt0 11360 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
57563ad2ant3 1104 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 0 < (𝑁 + 1))
5857adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 0 < (𝑁 + 1))
59 breq2 4689 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝑦) = (𝑁 + 1) → (0 < (#‘𝑦) ↔ 0 < (𝑁 + 1)))
6059ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (0 < (#‘𝑦) ↔ 0 < (𝑁 + 1)))
6158, 60mpbird 247 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 0 < (#‘𝑦))
62 hashle00 13226 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → ((#‘𝑦) ≤ 0 ↔ 𝑦 = ∅))
63 lencl 13356 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ Word 𝑉 → (#‘𝑦) ∈ ℕ0)
6463nn0red 11390 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ Word 𝑉 → (#‘𝑦) ∈ ℝ)
65 0re 10078 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
66 lenlt 10154 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((#‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((#‘𝑦) ≤ 0 ↔ ¬ 0 < (#‘𝑦)))
6766bicomd 213 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((#‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (¬ 0 < (#‘𝑦) ↔ (#‘𝑦) ≤ 0))
6864, 65, 67sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → (¬ 0 < (#‘𝑦) ↔ (#‘𝑦) ≤ 0))
69 nne 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
7069a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅))
7162, 68, 703bitr4rd 301 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (#‘𝑦)))
7271ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (#‘𝑦)))
7372con4bid 306 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ≠ ∅ ↔ 0 < (#‘𝑦)))
7461, 73mpbird 247 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑦 ≠ ∅)
7555, 74jca 553 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅))
7675ex 449 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
77763adant3 1101 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
7854, 77syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
7952, 78sylbi 207 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
8079imp 444 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅))
81 lswcl 13388 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Word 𝑉𝑦 ≠ ∅) → ( lastS ‘𝑦) ∈ 𝑉)
8280, 81syl 17 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ( lastS ‘𝑦) ∈ 𝑉)
8382ad2antrr 762 . . . . . . . . . . . . . 14 ((((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → ( lastS ‘𝑦) ∈ 𝑉)
84 preq1 4300 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ( lastS ‘𝑦) → {𝑝, 𝑛} = {( lastS ‘𝑦), 𝑛})
8584eleq1d 2715 . . . . . . . . . . . . . . . . . 18 (𝑝 = ( lastS ‘𝑦) → ({𝑝, 𝑛} ∈ (Edg‘𝐺) ↔ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)))
8685rabbidv 3220 . . . . . . . . . . . . . . . . 17 (𝑝 = ( lastS ‘𝑦) → {𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)} = {𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)})
8786fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑝 = ( lastS ‘𝑦) → (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
8887eqeq1d 2653 . . . . . . . . . . . . . . 15 (𝑝 = ( lastS ‘𝑦) → ((#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
8988rspcva 3338 . . . . . . . . . . . . . 14 ((( lastS ‘𝑦) ∈ 𝑉 ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
9083, 89sylancom 702 . . . . . . . . . . . . 13 ((((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
9190exp41 637 . . . . . . . . . . . 12 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))))
9291com14 96 . . . . . . . . . . 11 (∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))))
93923ad2ant3 1104 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))))
9449, 93syl 17 . . . . . . . . 9 (𝐺RegUSGraph𝐾 → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))))
9594imp41 618 . . . . . . . 8 ((((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
9644, 48, 953eqtrd 2689 . . . . . . 7 ((((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 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)
9796sumeq2dv 14477 . . . . . 6 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 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 6697 . . . . . . . 8 ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾𝑁) · 𝐾))
9998adantl 481 . . . . . . 7 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾𝑁) · 𝐾))
100 wwlksnfi 26869 . . . . . . . . . . . 12 ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
10129, 100sylbi 207 . . . . . . . . . . 11 (𝑉 ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
1021013ad2ant1 1102 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) ∈ Fin)
103102ad2antlr 763 . . . . . . . . 9 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝑁 WWalksN 𝐺) ∈ Fin)
104 rabfi 8226 . . . . . . . . 9 ((𝑁 WWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
105103, 104syl 17 . . . . . . . 8 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
106 rusgrusgr 26516 . . . . . . . . . . . . 13 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
107 simp1 1081 . . . . . . . . . . . . 13 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑉 ∈ Fin)
108106, 107anim12i 589 . . . . . . . . . . . 12 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1093isfusgr 26255 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
110108, 109sylibr 224 . . . . . . . . . . 11 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐺 ∈ FinUSGraph)
111 simpl 472 . . . . . . . . . . 11 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐺RegUSGraph𝐾)
112 ne0i 3954 . . . . . . . . . . . . 13 (𝑃𝑉𝑉 ≠ ∅)
1131123ad2ant2 1103 . . . . . . . . . . . 12 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑉 ≠ ∅)
114113adantl 481 . . . . . . . . . . 11 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑉 ≠ ∅)
1153frusgrnn0 26523 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
116110, 111, 114, 115syl3anc 1366 . . . . . . . . . 10 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐾 ∈ ℕ0)
117116nn0cnd 11391 . . . . . . . . 9 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐾 ∈ ℂ)
118117adantr 480 . . . . . . . 8 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 𝐾 ∈ ℂ)
119 fsumconst 14566 . . . . . . . 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))
120105, 118, 119syl2anc 694 . . . . . . 7 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾 = ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾))
121117, 2expp1d 13049 . . . . . . . 8 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
122121adantr 480 . . . . . . 7 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
12399, 120, 1223eqtr4d 2695 . . . . . 6 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾 = (𝐾↑(𝑁 + 1)))
12497, 123eqtrd 2685 . . . . 5 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 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)))
12516, 40, 1243eqtrd 2689 . . . 4 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1)))
126 peano2nn0 11371 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
1271263ad2ant3 1104 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
128127adantl 481 . . . . . 6 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑁 + 1) ∈ ℕ0)
1293, 4rusgrnumwwlklem 26937 . . . . . . 7 ((𝑃𝑉 ∧ (𝑁 + 1) ∈ ℕ0) → (𝑃𝐿(𝑁 + 1)) = (#‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
130129eqeq1d 2653 . . . . . 6 ((𝑃𝑉 ∧ (𝑁 + 1) ∈ ℕ0) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (#‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
1311, 128, 130syl2anc 694 . . . . 5 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (#‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
132131adantr 480 . . . 4 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (#‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
133125, 132mpbird 247 . . 3 (((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)))
134133ex 449 . 2 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
1357, 134sylbid 230 1 ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  c0 3948  {cpr 4212  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  cmpt2 6692  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  0cn0 11330  0*cxnn0 11401  ..^cfzo 12504  cexp 12900  #chash 13157  Word cword 13323   lastS clsw 13324   substr csubstr 13327  Σcsu 14460  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089  FinUSGraphcfusgr 26253  RegUSGraphcrusgr 26508   WWalksN cwwlksn 26774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-xadd 11985  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-ushgr 25999  df-upgr 26022  df-umgr 26023  df-uspgr 26090  df-usgr 26091  df-fusgr 26254  df-nbgr 26270  df-vtxdg 26418  df-rgr 26509  df-rusgr 26510  df-wwlks 26778  df-wwlksn 26779
This theorem is referenced by:  rusgrnumwwlk  26942
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