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Theorem rusgrnumwwlks 29772
Description: Induction step for rusgrnumwwlk 29773. (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 1193 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑃 ∈ 𝑉)
2 simpr3 1194 . . 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 29768 . . . 4 ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑃𝐿𝑁) = (β™―β€˜{𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}))
65eqeq1d 2729 . . 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 583 . 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 2727 . . . . . . . . . . . . 13 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
98wwlksnredwwlkn0 29694 . . . . . . . . . . . 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 412 . . . . . . . . . . 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 1133 . . . . . . . . . 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 481 . . . . . . . . 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 406 . . . . . . . 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 3434 . . . . . . 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 480 . . . . . 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 6895 . . . . 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 1135 . . . . . . . . . . . . 13 (((𝑀 prefix (𝑁 + 1)) = 𝑦 ∧ (π‘¦β€˜0) = 𝑃 ∧ {(lastSβ€˜π‘¦), (lastSβ€˜π‘€)} ∈ (Edgβ€˜πΊ)) β†’ (π‘¦β€˜0) = 𝑃)
1817pm4.71ri 560 . . . . . . . . . . . 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 3171 . . . . . . . . . 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 6890 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ (π‘₯β€˜0) = (π‘¦β€˜0))
2221eqeq1d 2729 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ ((π‘₯β€˜0) = 𝑃 ↔ (π‘¦β€˜0) = 𝑃))
2322rexrab 3689 . . . . . . . . . 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 3434 . . . . . . . 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 480 . . . . . . 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 6895 . . . . . 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 1213 . . . . . . 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 2819 . . . . . . . 8 (𝑉 ∈ Fin ↔ (Vtxβ€˜πΊ) ∈ Fin)
3029biimpi 215 . . . . . . 7 (𝑉 ∈ Fin β†’ (Vtxβ€˜πΊ) ∈ Fin)
31 eqid 2727 . . . . . . . 8 ((𝑁 + 1) WWalksN 𝐺) = ((𝑁 + 1) WWalksN 𝐺)
32 eqid 2727 . . . . . . . 8 {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} = {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃}
3331, 8, 32hashwwlksnext 29712 . . . . . . 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 6890 . . . . . . . . . 10 (π‘₯ = 𝑀 β†’ (π‘₯β€˜0) = (π‘€β€˜0))
3635eqeq1d 2729 . . . . . . . . 9 (π‘₯ = 𝑀 β†’ ((π‘₯β€˜0) = 𝑃 ↔ (π‘€β€˜0) = 𝑃))
3736cbvrabv 3437 . . . . . . . 8 {π‘₯ ∈ (𝑁 WWalksN 𝐺) ∣ (π‘₯β€˜0) = 𝑃} = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}
3837sumeq1i 15668 . . . . . . 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 2771 . . . . 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 29767 . . . . . . . . . . 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 2733 . . . . . . . . . 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 6895 . . . . . . . . 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 481 . . . . . . . 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 3674 . . . . . . . . . 10 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} β†’ 𝑦 ∈ (𝑁 WWalksN 𝐺))
4645adantl 481 . . . . . . . . 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 29701 . . . . . . . . 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 29386 . . . . . . . . . 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 6890 . . . . . . . . . . . . . . . . . . . 20 (𝑀 = 𝑦 β†’ (π‘€β€˜0) = (π‘¦β€˜0))
5150eqeq1d 2729 . . . . . . . . . . . . . . . . . . 19 (𝑀 = 𝑦 β†’ ((π‘€β€˜0) = 𝑃 ↔ (π‘¦β€˜0) = 𝑃))
5251elrab 3680 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃} ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (π‘¦β€˜0) = 𝑃))
533, 8wwlknp 29641 . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 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 12523 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ β„•0 β†’ 0 < (𝑁 + 1))
57563ad2ant3 1133 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 0 < (𝑁 + 1))
5857adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 0 < (𝑁 + 1))
59 breq2 5146 . . . . . . . . . . . . . . . . . . . . . . . . 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 14383 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 β†’ ((β™―β€˜π‘¦) ≀ 0 ↔ 𝑦 = βˆ…))
63 lencl 14507 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ Word 𝑉 β†’ (β™―β€˜π‘¦) ∈ β„•0)
6463nn0red 12555 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ Word 𝑉 β†’ (β™―β€˜π‘¦) ∈ ℝ)
65 0re 11238 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
66 lenlt 11314 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ Word 𝑉 β†’ (Β¬ 0 < (β™―β€˜π‘¦) ↔ (β™―β€˜π‘¦) ≀ 0))
69 nne 2939 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = (𝑁 + 1)) ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝑦 ∈ Word 𝑉 ∧ 𝑦 β‰  βˆ…))
7675ex 412 . . . . . . . . . . . . . . . . . . . 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 1130 . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . 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 14542 . . . . . . . . . . . . . . . 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 4733 . . . . . . . . . . . . . . . . . 18 (𝑝 = (lastSβ€˜π‘¦) β†’ {𝑝, 𝑛} = {(lastSβ€˜π‘¦), 𝑛})
8584eleq1d 2813 . . . . . . . . . . . . . . . . 17 (𝑝 = (lastSβ€˜π‘¦) β†’ ({𝑝, 𝑛} ∈ (Edgβ€˜πΊ) ↔ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)))
8685rabbidv 3435 . . . . . . . . . . . . . . . 16 (𝑝 = (lastSβ€˜π‘¦) β†’ {𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ (Edgβ€˜πΊ)} = {𝑛 ∈ 𝑉 ∣ {(lastSβ€˜π‘¦), 𝑛} ∈ (Edgβ€˜πΊ)})
8786fveqeq2d 6899 . . . . . . . . . . . . . . 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 3605 . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 1133 . . . . . . . . . 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 425 . . . . . . . 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 2771 . . . . . . 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 15673 . . . . . 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 7421 . . . . . . . 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 481 . . . . . . 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 29704 . . . . . . . . . . . 12 ((Vtxβ€˜πΊ) ∈ Fin β†’ (𝑁 WWalksN 𝐺) ∈ Fin)
10029, 99sylbi 216 . . . . . . . . . . 11 (𝑉 ∈ Fin β†’ (𝑁 WWalksN 𝐺) ∈ Fin)
1011003ad2ant1 1131 . . . . . . . . . 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 9285 . . . . . . . . 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 29365 . . . . . . . . . . . . 13 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USGraph)
106 simp1 1134 . . . . . . . . . . . . 13 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 𝑉 ∈ Fin)
107105, 106anim12i 612 . . . . . . . . . . . 12 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1083isfusgr 29118 . . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐺 RegUSGraph 𝐾)
111 ne0i 4330 . . . . . . . . . . . . 13 (𝑃 ∈ 𝑉 β†’ 𝑉 β‰  βˆ…)
1121113ad2ant2 1132 . . . . . . . . . . . 12 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ 𝑉 β‰  βˆ…)
113112adantl 481 . . . . . . . . . . 11 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝑉 β‰  βˆ…)
1143frusgrnn0 29372 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 β‰  βˆ…) β†’ 𝐾 ∈ β„•0)
115109, 110, 113, 114syl3anc 1369 . . . . . . . . . 10 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐾 ∈ β„•0)
116115nn0cnd 12556 . . . . . . . . 9 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ 𝐾 ∈ β„‚)
117116adantr 480 . . . . . . . 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 15760 . . . . . . . 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 583 . . . . . . 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 14135 . . . . . . . 8 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝐾↑(𝑁 + 1)) = ((𝐾↑𝑁) Β· 𝐾))
121120adantr 480 . . . . . . 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 2777 . . . . . 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 2767 . . . . 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 2771 . . . 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 12534 . . . . . . . 8 (𝑁 ∈ β„•0 β†’ (𝑁 + 1) ∈ β„•0)
1261253ad2ant3 1133 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (𝑁 + 1) ∈ β„•0)
127126adantl 481 . . . . . 6 ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ β„•0)) β†’ (𝑁 + 1) ∈ β„•0)
1283, 4rusgrnumwwlklem 29768 . . . . . . 7 ((𝑃 ∈ 𝑉 ∧ (𝑁 + 1) ∈ β„•0) β†’ (𝑃𝐿(𝑁 + 1)) = (β™―β€˜{𝑀 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑃}))
129128eqeq1d 2729 . . . . . 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 583 . . . . 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 480 . . . 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 412 . 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065  {crab 3427  βˆ…c0 4318  {cpr 4626   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  Fincfn 8955  β„‚cc 11128  β„cr 11129  0cc0 11130  1c1 11131   + caddc 11133   Β· cmul 11135   < clt 11270   ≀ cle 11271  β„•0cn0 12494  β„•0*cxnn0 12566  ..^cfzo 13651  β†‘cexp 14050  β™―chash 14313  Word cword 14488  lastSclsw 14536   prefix cpfx 14644  Ξ£csu 15656  Vtxcvtx 28796  Edgcedg 28847  USGraphcusgr 28949  FinUSGraphcfusgr 29116   RegUSGraph crusgr 29357   WWalksN cwwlksn 29624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-oi 9525  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-xadd 13117  df-fz 13509  df-fzo 13652  df-seq 13991  df-exp 14051  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-s1 14570  df-substr 14615  df-pfx 14645  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-vtx 28798  df-iedg 28799  df-edg 28848  df-uhgr 28858  df-ushgr 28859  df-upgr 28882  df-umgr 28883  df-uspgr 28950  df-usgr 28951  df-fusgr 29117  df-nbgr 29133  df-vtxdg 29267  df-rgr 29358  df-rusgr 29359  df-wwlks 29628  df-wwlksn 29629
This theorem is referenced by:  rusgrnumwwlk  29773
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