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Theorem rusgrnumwwlks 27680
Description: Induction step for rusgrnumwwlk 27681. (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 1187 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑃𝑉)
2 simpr3 1188 . . 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 27676 . . . 4 ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
65eqeq1d 2820 . . 3 ((𝑃𝑉𝑁 ∈ ℕ0) → ((𝑃𝐿𝑁) = (𝐾𝑁) ↔ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)))
71, 2, 6syl2anc 584 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) ↔ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)))
8 eqid 2818 . . . . . . . . . . . . 13 (Edg‘𝐺) = (Edg‘𝐺)
98wwlksnredwwlkn0 27601 . . . . . . . . . . . 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 413 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)))))
11103ad2ant3 1127 . . . . . . . . . 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 482 . . . . . . . . 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 407 . . . . . . . 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 3476 . . . . . . 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 481 . . . . . 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 6667 . . . . 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 1129 . . . . . . . . . . . . 13 (((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) → (𝑦‘0) = 𝑃)
1817pm4.71ri 561 . . . . . . . . . . . 12 (((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺)) ↔ ((𝑦‘0) = 𝑃 ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑤)} ∈ (Edg‘𝐺))))
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 3293 . . . . . . . . . 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 6662 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥‘0) = (𝑦‘0))
2221eqeq1d 2820 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
2322rexrab 3684 . . . . . . . . . 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, 23syl6bbr 290 . . . . . . . . 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 3476 . . . . . . . 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 481 . . . . . . 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 6667 . . . . . 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 1207 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 𝑉 ∈ Fin)
293eleq1i 2900 . . . . . . . 8 (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin)
3029biimpi 217 . . . . . . 7 (𝑉 ∈ Fin → (Vtx‘𝐺) ∈ Fin)
31 eqid 2818 . . . . . . . 8 ((𝑁 + 1) WWalksN 𝐺) = ((𝑁 + 1) WWalksN 𝐺)
32 eqid 2818 . . . . . . . 8 {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃}
3331, 8, 32hashwwlksnext 27620 . . . . . . 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 6662 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
3635eqeq1d 2820 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑥‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
3736cbvrabv 3489 . . . . . . . 8 {𝑥 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑥‘0) = 𝑃} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
3837sumeq1i 15043 . . . . . . 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 2857 . . . . 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 27675 . . . . . . . . . . 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 2824 . . . . . . . . . 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 6667 . . . . . . . . 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 482 . . . . . . . 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 3672 . . . . . . . . . 10 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → 𝑦 ∈ (𝑁 WWalksN 𝐺))
4645adantl 482 . . . . . . . . 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 27608 . . . . . . . . 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 27294 . . . . . . . . . 10 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑝𝑉 (♯‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
50 fveq1 6662 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
5150eqeq1d 2820 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
5251elrab 3677 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑦‘0) = 𝑃))
533, 8wwlknp 27548 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
5453adantr 481 . . . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑦 ∈ Word 𝑉)
56 nn0p1gt0 11914 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
57563ad2ant3 1127 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 0 < (𝑁 + 1))
5857adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 0 < (𝑁 + 1))
59 breq2 5061 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘𝑦) = (𝑁 + 1) → (0 < (♯‘𝑦) ↔ 0 < (𝑁 + 1)))
6059ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (0 < (♯‘𝑦) ↔ 0 < (𝑁 + 1)))
6158, 60mpbird 258 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 0 < (♯‘𝑦))
62 hashle00 13749 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → ((♯‘𝑦) ≤ 0 ↔ 𝑦 = ∅))
63 lencl 13871 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ Word 𝑉 → (♯‘𝑦) ∈ ℕ0)
6463nn0red 11944 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ Word 𝑉 → (♯‘𝑦) ∈ ℝ)
65 0re 10631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
66 lenlt 10707 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((♯‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → ((♯‘𝑦) ≤ 0 ↔ ¬ 0 < (♯‘𝑦)))
6766bicomd 224 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((♯‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (¬ 0 < (♯‘𝑦) ↔ (♯‘𝑦) ≤ 0))
6864, 65, 67sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → (¬ 0 < (♯‘𝑦) ↔ (♯‘𝑦) ≤ 0))
69 nne 3017 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
7069a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅))
7162, 68, 703bitr4rd 313 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (♯‘𝑦)))
7271ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (♯‘𝑦)))
7372con4bid 318 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ≠ ∅ ↔ 0 < (♯‘𝑦)))
7461, 73mpbird 258 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑦 ≠ ∅)
7555, 74jca 512 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅))
7675ex 413 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = (𝑁 + 1)) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
77763adant3 1124 . . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
8079imp 407 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅))
81 lswcl 13908 . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . 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 4661 . . . . . . . . . . . . . . . . . 18 (𝑝 = (lastS‘𝑦) → {𝑝, 𝑛} = {(lastS‘𝑦), 𝑛})
8584eleq1d 2894 . . . . . . . . . . . . . . . . 17 (𝑝 = (lastS‘𝑦) → ({𝑝, 𝑛} ∈ (Edg‘𝐺) ↔ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)))
8685rabbidv 3478 . . . . . . . . . . . . . . . 16 (𝑝 = (lastS‘𝑦) → {𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)} = {𝑛𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)})
8786fveqeq2d 6671 . . . . . . . . . . . . . . 15 (𝑝 = (lastS‘𝑦) → ((♯‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾 ↔ (♯‘{𝑛𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾))
8887rspcva 3618 . . . . . . . . . . . . . 14 (((lastS‘𝑦) ∈ 𝑉 ∧ ∀𝑝𝑉 (♯‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ (Edg‘𝐺)}) = 𝐾) → (♯‘{𝑛𝑉 ∣ {(lastS‘𝑦), 𝑛} ∈ (Edg‘𝐺)}) = 𝐾)
8983, 88sylancom 588 . . . . . . . . . . . . 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 435 . . . . . . . . . . . 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 1127 . . . . . . . . . 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 426 . . . . . . . 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 2857 . . . . . . 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 15048 . . . . . 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 7152 . . . . . . . 8 ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾𝑁) · 𝐾))
9897adantl 482 . . . . . . 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 27611 . . . . . . . . . . . 12 ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
10029, 99sylbi 218 . . . . . . . . . . 11 (𝑉 ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
1011003ad2ant1 1125 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) ∈ Fin)
102101ad2antlr 723 . . . . . . . . 9 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝑁 WWalksN 𝐺) ∈ Fin)
103 rabfi 8731 . . . . . . . . 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 27273 . . . . . . . . . . . . 13 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
106 simp1 1128 . . . . . . . . . . . . 13 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑉 ∈ Fin)
107105, 106anim12i 612 . . . . . . . . . . . 12 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1083isfusgr 27027 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
109107, 108sylibr 235 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐺 ∈ FinUSGraph)
110 simpl 483 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐺 RegUSGraph 𝐾)
111 ne0i 4297 . . . . . . . . . . . . 13 (𝑃𝑉𝑉 ≠ ∅)
1121113ad2ant2 1126 . . . . . . . . . . . 12 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑉 ≠ ∅)
113112adantl 482 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑉 ≠ ∅)
1143frusgrnn0 27280 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
115109, 110, 113, 114syl3anc 1363 . . . . . . . . . 10 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐾 ∈ ℕ0)
116115nn0cnd 11945 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝐾 ∈ ℂ)
117116adantr 481 . . . . . . . 8 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 𝐾 ∈ ℂ)
118 fsumconst 15133 . . . . . . . 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 584 . . . . . . 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 13499 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
121120adantr 481 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
12298, 119, 1213eqtr4d 2863 . . . . . 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 2853 . . . . 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 2857 . . . 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 11925 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
1261253ad2ant3 1127 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
127126adantl 482 . . . . . 6 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑁 + 1) ∈ ℕ0)
1283, 4rusgrnumwwlklem 27676 . . . . . . 7 ((𝑃𝑉 ∧ (𝑁 + 1) ∈ ℕ0) → (𝑃𝐿(𝑁 + 1)) = (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
129128eqeq1d 2820 . . . . . 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 584 . . . . 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 481 . . . 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 258 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)))
133132ex 413 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
1347, 133sylbid 241 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  c0 4288  {cpr 4559   class class class wbr 5057  cfv 6348  (class class class)co 7145  cmpo 7147  Fincfn 8497  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  0cn0 11885  0*cxnn0 11955  ..^cfzo 13021  cexp 13417  chash 13678  Word cword 13849  lastSclsw 13902   prefix cpfx 14020  Σcsu 15030  Vtxcvtx 26708  Edgcedg 26759  USGraphcusgr 26861  FinUSGraphcfusgr 27025   RegUSGraph crusgr 27265   WWalksN cwwlksn 27531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-xadd 12496  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-word 13850  df-lsw 13903  df-concat 13911  df-s1 13938  df-substr 13991  df-pfx 14021  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-vtx 26710  df-iedg 26711  df-edg 26760  df-uhgr 26770  df-ushgr 26771  df-upgr 26794  df-umgr 26795  df-uspgr 26862  df-usgr 26863  df-fusgr 27026  df-nbgr 27042  df-vtxdg 27175  df-rgr 27266  df-rusgr 27267  df-wwlks 27535  df-wwlksn 27536
This theorem is referenced by:  rusgrnumwwlk  27681
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