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Theorem rusgrnumwwlks 29805
Description: Induction step for rusgrnumwwlk 29806. (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 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
Assertion
Ref Expression
rusgrnumwwlks ((𝐺 RegUSGraph 𝐾 ∧ (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 1192 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑃𝑉)
2 simpr3 1193 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ (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 29801 . . . 4 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
65eqeq1d 2730 . . 3 ((𝑃𝑉𝑁 ∈ ℕ0) → ((𝑃𝐿𝑁) = (𝐾𝑁) ↔ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)))
71, 2, 6syl2anc 582 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) ↔ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)))
8 eqid 2728 . . . . . . . . . . . . 13 (Edg‘𝐺) = (Edg‘𝐺)
98wwlksnredwwlkn0 29727 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑤 ∈ ((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))))
109ex 411 . . . . . . . . . . 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)))))
11103ad2ant3 1132 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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)))))
1211adantl 480 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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)))))
1312imp 405 . . . . . . . 8 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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))))
1413rabbidva 3437 . . . . . . 7 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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))})
1514adantr 479 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (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 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))})
1615fveq2d 6906 . . . . 5 (((𝐺 RegUSGraph 𝐾 ∧ (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 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}))
17 simp2 1134 . . . . . . . . . . . . 13 (((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) → (𝑦‘0) = 𝑃)
1817pm4.71ri 559 . . . . . . . . . . . 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))))
1918a1i 11 . . . . . . . . . . 11 ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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)))))
2019rexbidva 3174 . . . . . . . . . 10 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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 6901 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥‘0) = (𝑦‘0))
2221eqeq1d 2730 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
2322rexrab 3693 . . . . . . . . . 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))))
2420, 23bitr4di 288 . . . . . . . . 9 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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))))
2524rabbidva 3437 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ 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))})
2625adantr 479 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → {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))})
2726fveq2d 6906 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (♯‘{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 1212 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 𝑉 ∈ Fin)
293eleq1i 2820 . . . . . . . 8 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
3029biimpi 215 . . . . . . 7 (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin)
31 eqid 2728 . . . . . . . 8 ((𝑁 + 1) WWalksN 𝐺) = ((𝑁 + 1) WWalksN 𝐺)
32 eqid 2728 . . . . . . . 8 {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃}
3331, 8, 32hashwwlksnext 29745 . . . . . . 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))}))
3428, 30, 333syl 18 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (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 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 6901 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
3635eqeq1d 2730 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑥‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
3736cbvrabv 3441 . . . . . . . 8 {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
3837sumeq1i 15684 . . . . . . 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))})
3938a1i 11 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (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 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))}))
4027, 34, 393eqtrd 2772 . . . . 5 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (♯‘{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 29800 . . . . . . . . . . 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))})
4241eqcomd 2734 . . . . . . . . . 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))})
4342fveq2d 6906 . . . . . . . . 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))}))
4443adantl 480 . . . . . . . 8 ((((𝐺 RegUSGraph 𝐾 ∧ (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 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 3678 . . . . . . . . . 10 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → 𝑦 ∈ (𝑁 WWalksN 𝐺))
4645adantl 480 . . . . . . . . 9 ((((𝐺 RegUSGraph 𝐾 ∧ (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 29734 . . . . . . . . 9 (𝑦 ∈ (𝑁 WWalksN 𝐺) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (♯‘{𝑛𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
4846, 47syl 17 . . . . . . . 8 ((((𝐺 RegUSGraph 𝐾 ∧ (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 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (♯‘{𝑛𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}))
493rusgrpropadjvtx 29419 . . . . . . . . . 10 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (♯‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
50 fveq1 6901 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
5150eqeq1d 2730 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
5251elrab 3684 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃))
533, 8wwlknp 29674 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5453adantr 479 . . . . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑦 ∈ Word 𝑉)
56 nn0p1gt0 12539 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
57563ad2ant3 1132 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 0 < (𝑁 + 1))
5857adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 0 < (𝑁 + 1))
59 breq2 5156 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘𝑦) = (𝑁 + 1) → (0 < (♯‘𝑦) ↔ 0 < (𝑁 + 1)))
6059ad2antlr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (0 < (♯‘𝑦) ↔ 0 < (𝑁 + 1)))
6158, 60mpbird 256 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 0 < (♯‘𝑦))
62 hashle00 14399 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → ((♯‘𝑦) ≤ 0 ↔ 𝑦 = ∅))
63 lencl 14523 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ Word 𝑉 → (♯‘𝑦) ∈ ℕ0)
6463nn0red 12571 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ Word 𝑉 → (♯‘𝑦) ∈ ℝ)
65 0re 11254 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
66 lenlt 11330 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((♯‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((♯‘𝑦) ≤ 0 ↔ ¬ 0 < (♯‘𝑦)))
6766bicomd 222 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((♯‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (¬ 0 < (♯‘𝑦) ↔ (♯‘𝑦) ≤ 0))
6864, 65, 67sylancl 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → (¬ 0 < (♯‘𝑦) ↔ (♯‘𝑦) ≤ 0))
69 nne 2941 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
7069a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅))
7162, 68, 703bitr4rd 311 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (♯‘𝑦)))
7271ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (♯‘𝑦)))
7372con4bid 316 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ≠ ∅ ↔ 0 < (♯‘𝑦)))
7461, 73mpbird 256 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑦 ≠ ∅)
7555, 74jca 510 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅))
7675ex 411 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
77763adant3 1129 . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
8079imp 405 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅))
81 lswcl 14558 . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 4742 . . . . . . . . . . . . . . . . . 18 (𝑝 = (lastS‘𝑦) → {𝑝, 𝑛} = {(lastS‘𝑦), 𝑛})
8584eleq1d 2814 . . . . . . . . . . . . . . . . 17 (𝑝 = (lastS‘𝑦) → ({𝑝, 𝑛} ∈ (Edg‘𝐺) ↔ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)))
8685rabbidv 3438 . . . . . . . . . . . . . . . 16 (𝑝 = (lastS‘𝑦) → {𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)} = {𝑛𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)})
8786fveqeq2d 6910 . . . . . . . . . . . . . . 15 (𝑝 = (lastS‘𝑦) → ((♯‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (♯‘{𝑛𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
8887rspcva 3609 . . . . . . . . . . . . . 14 (((lastS‘𝑦) ∈ 𝑉 ∧ ∀𝑝𝑉 (♯‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → (♯‘{𝑛𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
8983, 88sylancom 586 . . . . . . . . . . . . 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)
9089exp41 433 . . . . . . . . . . . 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))))
9190com14 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))))
92913ad2ant3 1132 . . . . . . . . . 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))))
9349, 92syl 17 . . . . . . . . 9 (𝐺 RegUSGraph 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))))
9493imp41 424 . . . . . . . 8 ((((𝐺 RegUSGraph 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)
9544, 48, 943eqtrd 2772 . . . . . . 7 ((((𝐺 RegUSGraph 𝐾 ∧ (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 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = 𝐾)
9695sumeq2dv 15689 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (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 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾)
97 oveq1 7433 . . . . . . . 8 ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾𝑁) · 𝐾))
9897adantl 480 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾𝑁) · 𝐾))
99 wwlksnfi 29737 . . . . . . . . . . . 12 ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
10029, 99sylbi 216 . . . . . . . . . . 11 (𝑉 ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
1011003ad2ant1 1130 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) ∈ Fin)
102101ad2antlr 725 . . . . . . . . 9 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝑁 WWalksN 𝐺) ∈ Fin)
103 rabfi 9300 . . . . . . . . 9 ((𝑁 WWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
104102, 103syl 17 . . . . . . . 8 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
105 rusgrusgr 29398 . . . . . . . . . . . . 13 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
106 simp1 1133 . . . . . . . . . . . . 13 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑉 ∈ Fin)
107105, 106anim12i 611 . . . . . . . . . . . 12 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1083isfusgr 29151 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
109107, 108sylibr 233 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐺 ∈ FinUSGraph)
110 simpl 481 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐺 RegUSGraph 𝐾)
111 ne0i 4338 . . . . . . . . . . . . 13 (𝑃𝑉𝑉 ≠ ∅)
1121113ad2ant2 1131 . . . . . . . . . . . 12 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑉 ≠ ∅)
113112adantl 480 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑉 ≠ ∅)
1143frusgrnn0 29405 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
115109, 110, 113, 114syl3anc 1368 . . . . . . . . . 10 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐾 ∈ ℕ0)
116115nn0cnd 12572 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐾 ∈ ℂ)
117116adantr 479 . . . . . . . 8 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 𝐾 ∈ ℂ)
118 fsumconst 15776 . . . . . . . 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))
119104, 117, 118syl2anc 582 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (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))
120116, 2expp1d 14151 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
121120adantr 479 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
12298, 119, 1213eqtr4d 2778 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → Σ𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}𝐾 = (𝐾↑(𝑁 + 1)))
12396, 122eqtrd 2768 . . . . 5 (((𝐺 RegUSGraph 𝐾 ∧ (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 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))}) = (𝐾↑(𝑁 + 1)))
12416, 40, 1233eqtrd 2772 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1)))
125 peano2nn0 12550 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
1261253ad2ant3 1132 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
127126adantl 480 . . . . . 6 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑁 + 1) ∈ ℕ0)
1283, 4rusgrnumwwlklem 29801 . . . . . . 7 ((𝑃𝑉 ∧ (𝑁 + 1) ∈ ℕ0) → (𝑃𝐿(𝑁 + 1)) = (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
129128eqeq1d 2730 . . . . . 6 ((𝑃𝑉 ∧ (𝑁 + 1) ∈ ℕ0) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
1301, 127, 129syl2anc 582 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
131130adantr 479 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ (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))))
132124, 131mpbird 256 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)))
133132ex 411 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
1347, 133sylbid 239 1 ((𝐺 RegUSGraph 𝐾 ∧ (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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2937  wral 3058  wrex 3067  {crab 3430  c0 4326  {cpr 4634   class class class wbr 5152  cfv 6553  (class class class)co 7426  cmpo 7428  Fincfn 8970  cc 11144  cr 11145  0cc0 11146  1c1 11147   + caddc 11149   · cmul 11151   < clt 11286  cle 11287  0cn0 12510  0*cxnn0 12582  ..^cfzo 13667  cexp 14066  chash 14329  Word cword 14504  lastSclsw 14552   prefix cpfx 14660  Σcsu 15672  Vtxcvtx 28829  Edgcedg 28880  USGraphcusgr 28982  FinUSGraphcfusgr 29149   RegUSGraph crusgr 29390   WWalksN cwwlksn 29657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-disj 5118  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-oadd 8497  df-er 8731  df-map 8853  df-pm 8854  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-oi 9541  df-dju 9932  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-xnn0 12583  df-z 12597  df-uz 12861  df-rp 13015  df-xadd 13133  df-fz 13525  df-fzo 13668  df-seq 14007  df-exp 14067  df-hash 14330  df-word 14505  df-lsw 14553  df-concat 14561  df-s1 14586  df-substr 14631  df-pfx 14661  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-sum 15673  df-vtx 28831  df-iedg 28832  df-edg 28881  df-uhgr 28891  df-ushgr 28892  df-upgr 28915  df-umgr 28916  df-uspgr 28983  df-usgr 28984  df-fusgr 29150  df-nbgr 29166  df-vtxdg 29300  df-rgr 29391  df-rusgr 29392  df-wwlks 29661  df-wwlksn 29662
This theorem is referenced by:  rusgrnumwwlk  29806
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