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Theorem rusgrnumwwlks 28961
Description: Induction step for rusgrnumwwlk 28962. (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 = (Vtxβ€˜πΊ)
rusgrnumwwlk.l 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„•0 ↦ (β™―β€˜{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}))
Assertion
Ref Expression
rusgrnumwwlks ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ ((𝑃𝐿𝑁) = (𝐾↑𝑁) β†’ (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑃,𝑛,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀   𝑀,𝐾
Allowed substitution hints:   𝐾(𝑣,𝑛)   𝐿(𝑀,𝑣,𝑛)

Proof of Theorem rusgrnumwwlks
Dummy variables 𝑖 𝑝 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr2 1196 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑃 ∈ 𝑉)
2 simpr3 1197 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑁 ∈ β„•0)
3 rusgrnumwwlk.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
4 rusgrnumwwlk.l . . . . 5 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„•0 ↦ (β™―β€˜{𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}))
53, 4rusgrnumwwlklem 28957 . . . 4 ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑃𝐿𝑁) = (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}))
65eqeq1d 2739 . . 3 ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((𝑃𝐿𝑁) = (𝐾↑𝑁) ↔ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)))
71, 2, 6syl2anc 585 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ ((𝑃𝐿𝑁) = (𝐾↑𝑁) ↔ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)))
8 eqid 2737 . . . . . . . . . . . . 13 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
98wwlksnredwwlkn0 28883 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ ((π‘€β€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))))
109ex 414 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ ((π‘€β€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)))))
11103ad2ant3 1136 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ ((π‘€β€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)))))
1211adantl 483 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) β†’ ((π‘€β€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)))))
1312imp 408 . . . . . . . 8 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ ((π‘€β€˜0) = 𝑃 ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))))
1413rabbidva 3417 . . . . . . 7 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))})
1514adantr 482 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))})
1615fveq2d 6851 . . . . 5 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}))
17 simp2 1138 . . . . . . . . . . . . 13 (((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)) β†’ (π‘¦β€˜0) = 𝑃)
1817pm4.71ri 562 . . . . . . . . . . . 12 (((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)) ↔ ((π‘¦β€˜0) = 𝑃 ∧ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))))
1918a1i 11 . . . . . . . . . . 11 ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺)) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) β†’ (((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)) ↔ ((π‘¦β€˜0) = 𝑃 ∧ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)))))
2019rexbidva 3174 . . . . . . . . . 10 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)) ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘¦β€˜0) = 𝑃 ∧ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)))))
21 fveq1 6846 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ (π‘₯β€˜0) = (π‘¦β€˜0))
2221eqeq1d 2739 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ ((π‘₯β€˜0) = 𝑃 ↔ (π‘¦β€˜0) = 𝑃))
2322rexrab 3659 . . . . . . . . . 10 (βˆƒπ‘¦ ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)) ↔ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((π‘¦β€˜0) = 𝑃 ∧ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))))
2420, 23bitr4di 289 . . . . . . . . 9 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ 𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺)) β†’ (βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)) ↔ βˆƒπ‘¦ ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))))
2524rabbidva 3417 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))} = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))})
2625adantr 482 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))} = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))})
2726fveq2d 6851 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}))
28 simplr1 1216 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ 𝑉 ∈ Fin)
293eleq1i 2829 . . . . . . . 8 (𝑉 ∈ Fin ↔ (Vtxβ€˜πΊ) ∈ Fin)
3029biimpi 215 . . . . . . 7 (𝑉 ∈ Fin β†’ (Vtxβ€˜πΊ) ∈ Fin)
31 eqid 2737 . . . . . . . 8 ((𝑁 + 1) WWalksN 𝐺) = ((𝑁 + 1) WWalksN 𝐺)
32 eqid 2737 . . . . . . . 8 {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} = {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃}
3331, 8, 32hashwwlksnext 28901 . . . . . . 7 ((Vtxβ€˜πΊ) ∈ Fin β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = Σ𝑦 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}))
3428, 30, 333syl 18 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = Σ𝑦 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}))
35 fveq1 6846 . . . . . . . . . 10 (π‘₯ = 𝑀 β†’ (π‘₯β€˜0) = (π‘€β€˜0))
3635eqeq1d 2739 . . . . . . . . 9 (π‘₯ = 𝑀 β†’ ((π‘₯β€˜0) = 𝑃 ↔ (π‘€β€˜0) = 𝑃))
3736cbvrabv 3420 . . . . . . . 8 {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}
3837sumeq1i 15590 . . . . . . 7 Σ𝑦 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))})
3938a1i 11 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ Σ𝑦 ∈ {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}))
4027, 34, 393eqtrd 2781 . . . . 5 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ βˆƒπ‘¦ ∈ (𝑁 WWalksN 𝐺)((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}))
41 rusgrnumwwlkslem 28956 . . . . . . . . . . 11 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))} = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))})
4241eqcomd 2743 . . . . . . . . . 10 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))} = {𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))})
4342fveq2d 6851 . . . . . . . . 9 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}))
4443adantl 483 . . . . . . . 8 ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}))
45 elrabi 3644 . . . . . . . . . 10 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ 𝑦 ∈ (𝑁 WWalksN 𝐺))
4645adantl 483 . . . . . . . . 9 ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) β†’ 𝑦 ∈ (𝑁 WWalksN 𝐺))
473, 8wwlksnexthasheq 28890 . . . . . . . . 9 (𝑦 ∈ (𝑁 WWalksN 𝐺) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}))
4846, 47syl 17 . . . . . . . 8 ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}))
493rusgrpropadjvtx 28575 . . . . . . . . . 10 (𝐺 RegUSGraph 𝐾 β†’ (𝐺 ∈ USGraph ∧ 𝐾 ∈ β„•0* ∧ βˆ€π‘ ∈ 𝑉 (β™―β€˜{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾))
50 fveq1 6846 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝑦 β†’ (π‘€β€˜0) = (π‘¦β€˜0))
5150eqeq1d 2739 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝑦 β†’ ((π‘€β€˜0) = 𝑃 ↔ (π‘¦β€˜0) = 𝑃))
5251elrab 3650 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (π‘¦β€˜0) = 𝑃))
533, 8wwlknp 28830 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑁 WWalksN 𝐺) β†’ (𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
5453adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (π‘¦β€˜0) = 𝑃) β†’ (𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
55 simpll 766 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑦 ∈ Word 𝑉)
56 nn0p1gt0 12449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„•0 β†’ 0 < (𝑁 + 1))
57563ad2ant3 1136 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 0 < (𝑁 + 1))
5857adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 0 < (𝑁 + 1))
59 breq2 5114 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β™―β€˜π‘¦) = (𝑁 + 1) β†’ (0 < (β™―β€˜π‘¦) ↔ 0 < (𝑁 + 1)))
6059ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (0 < (β™―β€˜π‘¦) ↔ 0 < (𝑁 + 1)))
6158, 60mpbird 257 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 0 < (β™―β€˜π‘¦))
62 hashle00 14307 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 β†’ ((β™―β€˜π‘¦) ≀ 0 ↔ 𝑦 = βˆ…))
63 lencl 14428 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ Word 𝑉 β†’ (β™―β€˜π‘¦) ∈ β„•0)
6463nn0red 12481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ Word 𝑉 β†’ (β™―β€˜π‘¦) ∈ ℝ)
65 0re 11164 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
66 lenlt 11240 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((β™―β€˜π‘¦) ∈ ℝ ∧ 0 ∈ ℝ) β†’ ((β™―β€˜π‘¦) ≀ 0 ↔ Β¬ 0 < (β™―β€˜π‘¦)))
6766bicomd 222 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((β™―β€˜π‘¦) ∈ ℝ ∧ 0 ∈ ℝ) β†’ (Β¬ 0 < (β™―β€˜π‘¦) ↔ (β™―β€˜π‘¦) ≀ 0))
6864, 65, 67sylancl 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 β†’ (Β¬ 0 < (β™―β€˜π‘¦) ↔ (β™―β€˜π‘¦) ≀ 0))
69 nne 2948 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Β¬ 𝑦 β‰  βˆ… ↔ 𝑦 = βˆ…)
7069a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 β†’ (Β¬ 𝑦 β‰  βˆ… ↔ 𝑦 = βˆ…))
7162, 68, 703bitr4rd 312 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ Word 𝑉 β†’ (Β¬ 𝑦 β‰  βˆ… ↔ Β¬ 0 < (β™―β€˜π‘¦)))
7271ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (Β¬ 𝑦 β‰  βˆ… ↔ Β¬ 0 < (β™―β€˜π‘¦)))
7372con4bid 317 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝑦 β‰  βˆ… ↔ 0 < (β™―β€˜π‘¦)))
7461, 73mpbird 257 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑦 β‰  βˆ…)
7555, 74jca 513 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝑦 ∈ Word 𝑉 ∧ 𝑦 β‰  βˆ…))
7675ex 414 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) β†’ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑦 ∈ Word 𝑉 ∧ 𝑦 β‰  βˆ…)))
77763adant3 1133 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑦 ∈ Word 𝑉 ∧ 𝑦 β‰  βˆ…)))
7854, 77syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (π‘¦β€˜0) = 𝑃) β†’ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑦 ∈ Word 𝑉 ∧ 𝑦 β‰  βˆ…)))
7952, 78sylbi 216 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑦 ∈ Word 𝑉 ∧ 𝑦 β‰  βˆ…)))
8079imp 408 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝑦 ∈ Word 𝑉 ∧ 𝑦 β‰  βˆ…))
81 lswcl 14463 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Word 𝑉 ∧ 𝑦 β‰  βˆ…) β†’ (lastSβ€˜π‘¦) ∈ 𝑉)
8280, 81syl 17 . . . . . . . . . . . . . . 15 ((𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (lastSβ€˜π‘¦) ∈ 𝑉)
8382ad2antrr 725 . . . . . . . . . . . . . 14 ((((𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) ∧ βˆ€π‘ ∈ 𝑉 (β™―β€˜{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾) β†’ (lastSβ€˜π‘¦) ∈ 𝑉)
84 preq1 4699 . . . . . . . . . . . . . . . . . 18 (𝑝 = (lastSβ€˜π‘¦) β†’ {𝑝, 𝑛} = {(lastSβ€˜π‘¦), 𝑛})
8584eleq1d 2823 . . . . . . . . . . . . . . . . 17 (𝑝 = (lastSβ€˜π‘¦) β†’ ({𝑝, 𝑛} ∈ (Edgβ€˜πΊ) ↔ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)))
8685rabbidv 3418 . . . . . . . . . . . . . . . 16 (𝑝 = (lastSβ€˜π‘¦) β†’ {𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)} = {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)})
8786fveqeq2d 6855 . . . . . . . . . . . . . . 15 (𝑝 = (lastSβ€˜π‘¦) β†’ ((β™―β€˜{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾 ↔ (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾))
8887rspcva 3582 . . . . . . . . . . . . . 14 (((lastSβ€˜π‘¦) ∈ 𝑉 ∧ βˆ€π‘ ∈ 𝑉 (β™―β€˜{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾) β†’ (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾)
8983, 88sylancom 589 . . . . . . . . . . . . 13 ((((𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) ∧ βˆ€π‘ ∈ 𝑉 (β™―β€˜{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾) β†’ (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾)
9089exp41 436 . . . . . . . . . . . 12 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁) β†’ (βˆ€π‘ ∈ 𝑉 (β™―β€˜{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾 β†’ (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾))))
9190com14 96 . . . . . . . . . . 11 (βˆ€π‘ ∈ 𝑉 (β™―β€˜{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾 β†’ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁) β†’ (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾))))
92913ad2ant3 1136 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ β„•0* ∧ βˆ€π‘ ∈ 𝑉 (β™―β€˜{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾) β†’ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁) β†’ (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾))))
9349, 92syl 17 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾 β†’ ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁) β†’ (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾))))
9493imp41 427 . . . . . . . 8 ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) β†’ (β™―β€˜{𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)}) = 𝐾)
9544, 48, 943eqtrd 2781 . . . . . . 7 ((((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) ∧ 𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = 𝐾)
9695sumeq2dv 15595 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}𝐾)
97 oveq1 7369 . . . . . . . 8 ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁) β†’ ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) Β· 𝐾) = ((𝐾↑𝑁) Β· 𝐾))
9897adantl 483 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) Β· 𝐾) = ((𝐾↑𝑁) Β· 𝐾))
99 wwlksnfi 28893 . . . . . . . . . . . 12 ((Vtxβ€˜πΊ) ∈ Fin β†’ (𝑁 WWalksN 𝐺) ∈ Fin)
10029, 99sylbi 216 . . . . . . . . . . 11 (𝑉 ∈ Fin β†’ (𝑁 WWalksN 𝐺) ∈ Fin)
1011003ad2ant1 1134 . . . . . . . . . 10 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑁 WWalksN 𝐺) ∈ Fin)
102101ad2antlr 726 . . . . . . . . 9 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ (𝑁 WWalksN 𝐺) ∈ Fin)
103 rabfi 9220 . . . . . . . . 9 ((𝑁 WWalksN 𝐺) ∈ Fin β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ∈ Fin)
104102, 103syl 17 . . . . . . . 8 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ∈ Fin)
105 rusgrusgr 28554 . . . . . . . . . . . . 13 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USGraph)
106 simp1 1137 . . . . . . . . . . . . 13 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 𝑉 ∈ Fin)
107105, 106anim12i 614 . . . . . . . . . . . 12 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1083isfusgr 28308 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
109107, 108sylibr 233 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐺 ∈ FinUSGraph)
110 simpl 484 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐺 RegUSGraph 𝐾)
111 ne0i 4299 . . . . . . . . . . . . 13 (𝑃 ∈ 𝑉 β†’ 𝑉 β‰  βˆ…)
1121113ad2ant2 1135 . . . . . . . . . . . 12 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 𝑉 β‰  βˆ…)
113112adantl 483 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑉 β‰  βˆ…)
1143frusgrnn0 28561 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 β‰  βˆ…) β†’ 𝐾 ∈ β„•0)
115109, 110, 113, 114syl3anc 1372 . . . . . . . . . 10 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐾 ∈ β„•0)
116115nn0cnd 12482 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐾 ∈ β„‚)
117116adantr 482 . . . . . . . 8 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ 𝐾 ∈ β„‚)
118 fsumconst 15682 . . . . . . . 8 (({𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ∈ Fin ∧ 𝐾 ∈ β„‚) β†’ Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}𝐾 = ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) Β· 𝐾))
119104, 117, 118syl2anc 585 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}𝐾 = ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) Β· 𝐾))
120116, 2expp1d 14059 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝐾↑(𝑁 + 1)) = ((𝐾↑𝑁) Β· 𝐾))
121120adantr 482 . . . . . . 7 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ (𝐾↑(𝑁 + 1)) = ((𝐾↑𝑁) Β· 𝐾))
12298, 119, 1213eqtr4d 2787 . . . . . 6 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}𝐾 = (𝐾↑(𝑁 + 1)))
12396, 122eqtrd 2777 . . . . 5 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ Σ𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ))}) = (𝐾↑(𝑁 + 1)))
12416, 40, 1233eqtrd 2781 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑(𝑁 + 1)))
125 peano2nn0 12460 . . . . . . . 8 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
1261253ad2ant3 1136 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑁 + 1) ∈ β„•0)
127126adantl 483 . . . . . 6 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝑁 + 1) ∈ β„•0)
1283, 4rusgrnumwwlklem 28957 . . . . . . 7 ((𝑃 ∈ 𝑉 ∧ (𝑁 + 1) ∈ β„•0) β†’ (𝑃𝐿(𝑁 + 1)) = (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}))
129128eqeq1d 2739 . . . . . 6 ((𝑃 ∈ 𝑉 ∧ (𝑁 + 1) ∈ β„•0) β†’ ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
1301, 127, 129syl2anc 585 . . . . 5 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
131130adantr 482 . . . 4 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ ((𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)) ↔ (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑(𝑁 + 1))))
132124, 131mpbird 257 . . 3 (((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) ∧ (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁)) β†’ (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)))
133132ex 414 . 2 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ ((β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}) = (𝐾↑𝑁) β†’ (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
1347, 133sylbid 239 1 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ ((𝑃𝐿𝑁) = (𝐾↑𝑁) β†’ (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  βˆ…c0 4287  {cpr 4593   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  Fincfn 8890  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   Β· cmul 11063   < clt 11196   ≀ cle 11197  β„•0cn0 12420  β„•0*cxnn0 12492  ..^cfzo 13574  β†‘cexp 13974  β™―chash 14237  Word cword 14409  lastSclsw 14457   prefix cpfx 14565  Ξ£csu 15577  Vtxcvtx 27989  Edgcedg 28040  USGraphcusgr 28142  FinUSGraphcfusgr 28306   RegUSGraph crusgr 28546   WWalksN cwwlksn 28813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-oi 9453  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-rp 12923  df-xadd 13041  df-fz 13432  df-fzo 13575  df-seq 13914  df-exp 13975  df-hash 14238  df-word 14410  df-lsw 14458  df-concat 14466  df-s1 14491  df-substr 14536  df-pfx 14566  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-sum 15578  df-vtx 27991  df-iedg 27992  df-edg 28041  df-uhgr 28051  df-ushgr 28052  df-upgr 28075  df-umgr 28076  df-uspgr 28143  df-usgr 28144  df-fusgr 28307  df-nbgr 28323  df-vtxdg 28456  df-rgr 28547  df-rusgr 28548  df-wwlks 28817  df-wwlksn 28818
This theorem is referenced by:  rusgrnumwwlk  28962
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