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Theorem rusgranumwlks 26483
 Description: Induction step for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
Hypotheses
Ref Expression
rusgranumwlk.w 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
rusgranumwlk.l 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
Assertion
Ref Expression
rusgranumwlks ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
Distinct variable groups:   𝐸,𝑐,𝑛   𝑁,𝑐,𝑛   𝑉,𝑐,𝑛   𝑣,𝑁,𝑤   𝑃,𝑛,𝑣,𝑤   𝑣,𝑉   𝑛,𝑊,𝑣,𝑤   𝑤,𝑉,𝑐   𝑣,𝐸,𝑤   𝑤,𝐾
Allowed substitution hints:   𝑃(𝑐)   𝐾(𝑣,𝑛,𝑐)   𝐿(𝑤,𝑣,𝑛,𝑐)   𝑊(𝑐)

Proof of Theorem rusgranumwlks
Dummy variables 𝑖 𝑝 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rusisusgra 26458 . . . 4 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)
21adantr 480 . . 3 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑉 USGrph 𝐸)
3 simpr2 1061 . . 3 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑃𝑉)
4 simp3 1056 . . . 4 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
54adantl 481 . . 3 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑁 ∈ ℕ0)
6 rusgranumwlk.w . . . . 5 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})
7 rusgranumwlk.l . . . . 5 𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))
86, 7rusgranumwlklem4 26479 . . . 4 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
98eqeq1d 2612 . . 3 ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → ((𝑃𝐿𝑁) = (𝐾𝑁) ↔ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)))
102, 3, 5, 9syl3anc 1318 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) ↔ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)))
11 wwlknredwwlkn0 26255 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)))
1211ex 449 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸))))
13123ad2ant3 1077 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸))))
1413adantl 481 . . . . . . . . 9 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸))))
1514imp 444 . . . . . . . 8 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑤‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)))
1615rabbidva 3163 . . . . . . 7 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)})
1716adantr 480 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)})
1817fveq2d 6107 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ (𝑤‘0) = 𝑃}) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}))
19 simp2 1055 . . . . . . . . . . . . 13 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸) → (𝑦‘0) = 𝑃)
2019pm4.71ri 663 . . . . . . . . . . . 12 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ((𝑦‘0) = 𝑃 ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)))
2120a1i 11 . . . . . . . . . . 11 ((((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ((𝑦‘0) = 𝑃 ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸))))
2221rexbidva 3031 . . . . . . . . . 10 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑦‘0) = 𝑃 ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸))))
23 fveq1 6102 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥‘0) = (𝑦‘0))
2423eqeq1d 2612 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
2524rexrab 3337 . . . . . . . . . 10 (∃𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑦‘0) = 𝑃 ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)))
2622, 25syl6bbr 277 . . . . . . . . 9 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ∃𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)))
2726rabbidva 3163 . . . . . . . 8 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)})
2827adantr 480 . . . . . . 7 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)})
2928fveq2d 6107 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}))
30 simplr1 1096 . . . . . . 7 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 𝑉 ∈ Fin)
31 eqid 2610 . . . . . . . 8 ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))
32 eqid 2610 . . . . . . . 8 {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃}
3331, 32hashwwlkext 26274 . . . . . . 7 (𝑉 ∈ Fin → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = Σ𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}))
3430, 33syl 17 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = Σ𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}))
35 fveq1 6102 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑥‘0) = (𝑤‘0))
3635eqeq1d 2612 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑥‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
3736cbvrabv 3172 . . . . . . . 8 {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
3837sumeq1i 14276 . . . . . . 7 Σ𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)})
3938a1i 11 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → Σ𝑦 ∈ {𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑥‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}))
4029, 34, 393eqtrd 2648 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}))
41 rusgranumwlklem0 26475 . . . . . . . . . . 11 (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)})
4241eqcomd 2616 . . . . . . . . . 10 (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)})
4342fveq2d 6107 . . . . . . . . 9 (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}))
4443adantl 481 . . . . . . . 8 ((((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}))
45 elrabi 3328 . . . . . . . . . 10 (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
4645adantl 481 . . . . . . . . 9 ((((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) → 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
47 wwlkexthasheq 26262 . . . . . . . . 9 (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}))
4846, 47syl 17 . . . . . . . 8 ((((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}))
49 rusgraprop3 26470 . . . . . . . . . 10 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾))
50 fveq1 6102 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
5150eqeq1d 2612 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑦 → ((𝑤‘0) = 𝑃 ↔ (𝑦‘0) = 𝑃))
5251elrab 3331 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} ↔ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑦‘0) = 𝑃))
53 wwlknimp 26215 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ ran 𝐸))
5453adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑦‘0) = 𝑃) → (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ ran 𝐸))
55 simpll 786 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 𝑦 ∈ Word 𝑉)
56 nn0p1gt0 11199 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 1))
57563ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . . . . . 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 4587 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝑦) = (𝑁 + 1) → (0 < (#‘𝑦) ↔ 0 < (𝑁 + 1)))
6059ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (0 < (#‘𝑦) ↔ 0 < (𝑁 + 1)))
6158, 60mpbird 246 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → 0 < (#‘𝑦))
62 hashle00 13049 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → ((#‘𝑦) ≤ 0 ↔ 𝑦 = ∅))
63 lencl 13179 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ Word 𝑉 → (#‘𝑦) ∈ ℕ0)
6463nn0red 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)))
6766bicomd 212 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((#‘𝑦) ∈ ℝ ∧ 0 ∈ ℝ) → (¬ 0 < (#‘𝑦) ↔ (#‘𝑦) ≤ 0))
6864, 65, 67sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → (¬ 0 < (#‘𝑦) ↔ (#‘𝑦) ≤ 0))
69 nne 2786 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
7069a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ 𝑦 = ∅))
7162, 68, 703bitr4rd 300 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ Word 𝑉 → (¬ 𝑦 ≠ ∅ ↔ ¬ 0 < (#‘𝑦)))
7271ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
7854, 77syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑦‘0) = 𝑃) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
7952, 78sylbi 206 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅)))
8079imp 444 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑦 ∈ Word 𝑉𝑦 ≠ ∅))
81 lswcl 13208 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Word 𝑉𝑦 ≠ ∅) → ( lastS ‘𝑦) ∈ 𝑉)
8280, 81syl 17 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ( lastS ‘𝑦) ∈ 𝑉)
8382ad2antrr 758 . . . . . . . . . . . . . 14 ((((𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾) → ( lastS ‘𝑦) ∈ 𝑉)
84 preq1 4212 . . . . . . . . . . . . . . . . . . 19 (𝑝 = ( lastS ‘𝑦) → {𝑝, 𝑛} = {( lastS ‘𝑦), 𝑛})
8584eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑝 = ( lastS ‘𝑦) → ({𝑝, 𝑛} ∈ ran 𝐸 ↔ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸))
8685rabbidv 3164 . . . . . . . . . . . . . . . . 17 (𝑝 = ( lastS ‘𝑦) → {𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸} = {𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸})
8786fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑝 = ( lastS ‘𝑦) → (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}))
8887eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑝 = ( lastS ‘𝑦) → ((#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}) = 𝐾))
8988rspcva 3280 . . . . . . . . . . . . . 14 ((( lastS ‘𝑦) ∈ 𝑉 ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾) → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}) = 𝐾)
9083, 89sylancom 698 . . . . . . . . . . . . 13 ((((𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾) → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}) = 𝐾)
9190exp41 636 . . . . . . . . . . . 12 (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾 → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}) = 𝐾))))
9291com14 94 . . . . . . . . . . 11 (∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾 → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}) = 𝐾))))
93923ad2ant3 1077 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾) → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}) = 𝐾))))
9449, 93syl 17 . . . . . . . . 9 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}) = 𝐾))))
9594imp41 617 . . . . . . . 8 ((((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑦), 𝑛} ∈ ran 𝐸}) = 𝐾)
9644, 48, 953eqtrd 2648 . . . . . . 7 ((((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) ∧ 𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = 𝐾)
9796sumeq2dv 14281 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}𝐾)
98 oveq1 6556 . . . . . . . 8 ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾𝑁) · 𝐾))
9998adantl 481 . . . . . . 7 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) · 𝐾) = ((𝐾𝑁) · 𝐾))
100 wwlknfi 26266 . . . . . . . . . . 11 (𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)
1011003ad2ant1 1075 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)
102101ad2antlr 759 . . . . . . . . 9 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)
103 rabfi 8070 . . . . . . . . 9 (((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
104102, 103syl 17 . . . . . . . 8 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} ∈ Fin)
105 rusgraprop 26456 . . . . . . . . . 10 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
106 nn0cn 11179 . . . . . . . . . . 11 (𝐾 ∈ ℕ0𝐾 ∈ ℂ)
1071063ad2ant2 1076 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾) → 𝐾 ∈ ℂ)
108105, 107syl 17 . . . . . . . . 9 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝐾 ∈ ℂ)
109108ad2antrr 758 . . . . . . . 8 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → 𝐾 ∈ ℂ)
110 fsumconst 14364 . . . . . . . 8 (({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} ∈ Fin ∧ 𝐾 ∈ ℂ) → Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}𝐾 = ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) · 𝐾))
111104, 109, 110syl2anc 691 . . . . . . 7 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}𝐾 = ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) · 𝐾))
112 expp1 12729 . . . . . . . . 9 ((𝐾 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
113108, 4, 112syl2an 493 . . . . . . . 8 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
114113adantr 480 . . . . . . 7 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝐾↑(𝑁 + 1)) = ((𝐾𝑁) · 𝐾))
11599, 111, 1143eqtr4d 2654 . . . . . 6 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}𝐾 = (𝐾↑(𝑁 + 1)))
11697, 115eqtrd 2644 . . . . 5 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → Σ𝑦 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = (𝐾↑(𝑁 + 1)))
11718, 40, 1163eqtrd 2648 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1)))
118 peano2nn0 11210 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
1191183ad2ant3 1077 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
120119adantl 481 . . . . . 6 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑁 + 1) ∈ ℕ0)
1216, 7rusgranumwlklem4 26479 . . . . . . 7 ((𝑉 USGrph 𝐸𝑃𝑉 ∧ (𝑁 + 1) ∈ ℕ0) → (𝑃𝐿(𝑁 + 1)) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ (𝑤‘0) = 𝑃}))
122121eqeq1d 2612 . . . . . 6 ((𝑉 USGrph 𝐸𝑃𝑉 ∧ (𝑁 + 1) ∈ ℕ0) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
1232, 3, 120, 122syl3anc 1318 . . . . 5 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
124123adantr 480 . . . 4 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
125117, 124mpbird 246 . . 3 (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) ∧ (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁)) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)))
126125ex 449 . 2 ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
12710, 126sylbid 229 1 ((⟨𝑉, 𝐸⟩ RegUSGrph 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 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   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954  ℕ0cn0 11169  ..^cfzo 12334  ↑cexp 12722  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150  Σcsu 14264   USGrph cusg 25859   Walks cwalk 26026   WWalksN cwwlkn 26206   VDeg cvdg 26420   RegUSGrph crusgra 26450 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-usgra 25862  df-nbgra 25949  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  rusgranumwlk  26484
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