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Theorem numclwwlk1lem2f1 29343
Description: 𝑇 is a 1-1 function. (Contributed by AV, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 23-Feb-2022.) (Revised by AV, 31-Oct-2022.)
Hypotheses
Ref Expression
extwwlkfab.v 𝑉 = (Vtxβ€˜πΊ)
extwwlkfab.c 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
extwwlkfab.f 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))
numclwwlk.t 𝑇 = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)
Assertion
Ref Expression
numclwwlk1lem2f1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)–1-1β†’(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀   𝑛,𝑋,𝑣,𝑀   𝑀,𝐹   𝑒,𝐢   𝑒,𝐹   𝑒,𝐺,𝑀   𝑒,𝑁   𝑒,𝑉   𝑒,𝑋   𝑒,𝑇
Allowed substitution hints:   𝐢(𝑀,𝑣,𝑛)   𝑇(𝑀,𝑣,𝑛)   𝐹(𝑣,𝑛)

Proof of Theorem numclwwlk1lem2f1
Dummy variables π‘Ž 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 extwwlkfab.v . . 3 𝑉 = (Vtxβ€˜πΊ)
2 extwwlkfab.c . . 3 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
3 extwwlkfab.f . . 3 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))
4 numclwwlk.t . . 3 𝑇 = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)
51, 2, 3, 4numclwwlk1lem2f 29341 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)⟢(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
61, 2, 3, 4numclwwlk1lem2fv 29342 . . . . . 6 (𝑝 ∈ (𝑋𝐢𝑁) β†’ (π‘‡β€˜π‘) = ⟨(𝑝 prefix (𝑁 βˆ’ 2)), (π‘β€˜(𝑁 βˆ’ 1))⟩)
76ad2antrl 727 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁))) β†’ (π‘‡β€˜π‘) = ⟨(𝑝 prefix (𝑁 βˆ’ 2)), (π‘β€˜(𝑁 βˆ’ 1))⟩)
81, 2, 3, 4numclwwlk1lem2fv 29342 . . . . . 6 (π‘Ž ∈ (𝑋𝐢𝑁) β†’ (π‘‡β€˜π‘Ž) = ⟨(π‘Ž prefix (𝑁 βˆ’ 2)), (π‘Žβ€˜(𝑁 βˆ’ 1))⟩)
98ad2antll 728 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁))) β†’ (π‘‡β€˜π‘Ž) = ⟨(π‘Ž prefix (𝑁 βˆ’ 2)), (π‘Žβ€˜(𝑁 βˆ’ 1))⟩)
107, 9eqeq12d 2753 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁))) β†’ ((π‘‡β€˜π‘) = (π‘‡β€˜π‘Ž) ↔ ⟨(𝑝 prefix (𝑁 βˆ’ 2)), (π‘β€˜(𝑁 βˆ’ 1))⟩ = ⟨(π‘Ž prefix (𝑁 βˆ’ 2)), (π‘Žβ€˜(𝑁 βˆ’ 1))⟩))
11 ovex 7395 . . . . . 6 (𝑝 prefix (𝑁 βˆ’ 2)) ∈ V
12 fvex 6860 . . . . . 6 (π‘β€˜(𝑁 βˆ’ 1)) ∈ V
1311, 12opth 5438 . . . . 5 (⟨(𝑝 prefix (𝑁 βˆ’ 2)), (π‘β€˜(𝑁 βˆ’ 1))⟩ = ⟨(π‘Ž prefix (𝑁 βˆ’ 2)), (π‘Žβ€˜(𝑁 βˆ’ 1))⟩ ↔ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1))))
14 uzuzle23 12821 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
1522clwwlkel 29335 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑝 ∈ (𝑋𝐢𝑁) ↔ (𝑝 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋)))
16 isclwwlknon 29077 . . . . . . . . . . . 12 (𝑝 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ (𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋))
1716anbi1i 625 . . . . . . . . . . 11 ((𝑝 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋))
1815, 17bitrdi 287 . . . . . . . . . 10 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑝 ∈ (𝑋𝐢𝑁) ↔ ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋)))
1922clwwlkel 29335 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (π‘Ž ∈ (𝑋𝐢𝑁) ↔ (π‘Ž ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)))
20 isclwwlknon 29077 . . . . . . . . . . . 12 (π‘Ž ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋))
2120anbi1i 625 . . . . . . . . . . 11 ((π‘Ž ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋))
2219, 21bitrdi 287 . . . . . . . . . 10 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (π‘Ž ∈ (𝑋𝐢𝑁) ↔ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)))
2318, 22anbi12d 632 . . . . . . . . 9 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ ((𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁)) ↔ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋))))
2414, 23sylan2 594 . . . . . . . 8 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁)) ↔ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋))))
25243adant1 1131 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁)) ↔ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋))))
261clwwlknbp 29021 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝑁 ClWWalksN 𝐺) β†’ (𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁))
2726adantr 482 . . . . . . . . . . . . . 14 ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) β†’ (𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁))
2827adantr 482 . . . . . . . . . . . . 13 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁))
29 simpr 486 . . . . . . . . . . . . . 14 ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) β†’ (π‘β€˜0) = 𝑋)
3029adantr 482 . . . . . . . . . . . . 13 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘β€˜0) = 𝑋)
31 simpr 486 . . . . . . . . . . . . . 14 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋)
3229eqcomd 2743 . . . . . . . . . . . . . . 15 ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) β†’ 𝑋 = (π‘β€˜0))
3332adantr 482 . . . . . . . . . . . . . 14 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑋 = (π‘β€˜0))
3431, 33eqtrd 2777 . . . . . . . . . . . . 13 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘β€˜(𝑁 βˆ’ 2)) = (π‘β€˜0))
3528, 30, 34jca32 517 . . . . . . . . . . . 12 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁) ∧ ((π‘β€˜0) = 𝑋 ∧ (π‘β€˜(𝑁 βˆ’ 2)) = (π‘β€˜0))))
361clwwlknbp 29021 . . . . . . . . . . . . . . 15 (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) β†’ (π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁))
3736adantr 482 . . . . . . . . . . . . . 14 ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) β†’ (π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁))
3837adantr 482 . . . . . . . . . . . . 13 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁))
39 simpr 486 . . . . . . . . . . . . . 14 ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) β†’ (π‘Žβ€˜0) = 𝑋)
4039adantr 482 . . . . . . . . . . . . 13 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘Žβ€˜0) = 𝑋)
41 simpr 486 . . . . . . . . . . . . . 14 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)
4239eqcomd 2743 . . . . . . . . . . . . . . 15 ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) β†’ 𝑋 = (π‘Žβ€˜0))
4342adantr 482 . . . . . . . . . . . . . 14 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑋 = (π‘Žβ€˜0))
4441, 43eqtrd 2777 . . . . . . . . . . . . 13 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘Žβ€˜(𝑁 βˆ’ 2)) = (π‘Žβ€˜0))
4538, 40, 44jca32 517 . . . . . . . . . . . 12 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) ∧ ((π‘Žβ€˜0) = 𝑋 ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = (π‘Žβ€˜0))))
46 eqtr3 2763 . . . . . . . . . . . . . . . . 17 (((β™―β€˜π‘) = 𝑁 ∧ (β™―β€˜π‘Ž) = 𝑁) β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž))
4746expcom 415 . . . . . . . . . . . . . . . 16 ((β™―β€˜π‘Ž) = 𝑁 β†’ ((β™―β€˜π‘) = 𝑁 β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž)))
4847ad2antlr 726 . . . . . . . . . . . . . . 15 (((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) ∧ ((π‘Žβ€˜0) = 𝑋 ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = (π‘Žβ€˜0))) β†’ ((β™―β€˜π‘) = 𝑁 β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž)))
4948com12 32 . . . . . . . . . . . . . 14 ((β™―β€˜π‘) = 𝑁 β†’ (((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) ∧ ((π‘Žβ€˜0) = 𝑋 ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = (π‘Žβ€˜0))) β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž)))
5049ad2antlr 726 . . . . . . . . . . . . 13 (((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁) ∧ ((π‘β€˜0) = 𝑋 ∧ (π‘β€˜(𝑁 βˆ’ 2)) = (π‘β€˜0))) β†’ (((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) ∧ ((π‘Žβ€˜0) = 𝑋 ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = (π‘Žβ€˜0))) β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž)))
5150imp 408 . . . . . . . . . . . 12 ((((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁) ∧ ((π‘β€˜0) = 𝑋 ∧ (π‘β€˜(𝑁 βˆ’ 2)) = (π‘β€˜0))) ∧ ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) ∧ ((π‘Žβ€˜0) = 𝑋 ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = (π‘Žβ€˜0)))) β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž))
5235, 45, 51syl2an 597 . . . . . . . . . . 11 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž))
53523ad2ant2 1135 . . . . . . . . . 10 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž))
5427simprd 497 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) β†’ (β™―β€˜π‘) = 𝑁)
5554adantr 482 . . . . . . . . . . . . . . . . . . . 20 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (β™―β€˜π‘) = 𝑁)
5655eqcomd 2743 . . . . . . . . . . . . . . . . . . 19 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑁 = (β™―β€˜π‘))
5756adantr 482 . . . . . . . . . . . . . . . . . 18 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ 𝑁 = (β™―β€˜π‘))
5857oveq1d 7377 . . . . . . . . . . . . . . . . 17 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (𝑁 βˆ’ 2) = ((β™―β€˜π‘) βˆ’ 2))
5958oveq2d 7378 . . . . . . . . . . . . . . . 16 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (𝑝 prefix (𝑁 βˆ’ 2)) = (𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)))
6058oveq2d 7378 . . . . . . . . . . . . . . . 16 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘Ž prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2)))
6159, 60eqeq12d 2753 . . . . . . . . . . . . . . 15 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ↔ (𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2))))
6261biimpcd 249 . . . . . . . . . . . . . 14 ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) β†’ ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2))))
6362adantr 482 . . . . . . . . . . . . 13 (((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1))) β†’ ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2))))
6463impcom 409 . . . . . . . . . . . 12 (((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ (𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2)))
6555oveq1d 7377 . . . . . . . . . . . . . . . . 17 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ ((β™―β€˜π‘) βˆ’ 2) = (𝑁 βˆ’ 2))
6665fveq2d 6851 . . . . . . . . . . . . . . . 16 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = (π‘β€˜(𝑁 βˆ’ 2)))
6766, 31eqtrd 2777 . . . . . . . . . . . . . . 15 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = 𝑋)
6867adantr 482 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = 𝑋)
6941eqcomd 2743 . . . . . . . . . . . . . . . 16 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑋 = (π‘Žβ€˜(𝑁 βˆ’ 2)))
7069adantl 483 . . . . . . . . . . . . . . 15 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ 𝑋 = (π‘Žβ€˜(𝑁 βˆ’ 2)))
7158fveq2d 6851 . . . . . . . . . . . . . . 15 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘Žβ€˜(𝑁 βˆ’ 2)) = (π‘Žβ€˜((β™―β€˜π‘) βˆ’ 2)))
7270, 71eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ 𝑋 = (π‘Žβ€˜((β™―β€˜π‘) βˆ’ 2)))
7368, 72eqtrd 2777 . . . . . . . . . . . . 13 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = (π‘Žβ€˜((β™―β€˜π‘) βˆ’ 2)))
7473adantr 482 . . . . . . . . . . . 12 (((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = (π‘Žβ€˜((β™―β€˜π‘) βˆ’ 2)))
75 lsw 14459 . . . . . . . . . . . . . . . . . . . 20 (𝑝 ∈ Word 𝑉 β†’ (lastSβ€˜π‘) = (π‘β€˜((β™―β€˜π‘) βˆ’ 1)))
76 fvoveq1 7385 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘) = 𝑁 β†’ (π‘β€˜((β™―β€˜π‘) βˆ’ 1)) = (π‘β€˜(𝑁 βˆ’ 1)))
7775, 76sylan9eq 2797 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁) β†’ (lastSβ€˜π‘) = (π‘β€˜(𝑁 βˆ’ 1)))
7826, 77syl 17 . . . . . . . . . . . . . . . . . 18 (𝑝 ∈ (𝑁 ClWWalksN 𝐺) β†’ (lastSβ€˜π‘) = (π‘β€˜(𝑁 βˆ’ 1)))
7978eqcomd 2743 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ (𝑁 ClWWalksN 𝐺) β†’ (π‘β€˜(𝑁 βˆ’ 1)) = (lastSβ€˜π‘))
8079ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘β€˜(𝑁 βˆ’ 1)) = (lastSβ€˜π‘))
81 lsw 14459 . . . . . . . . . . . . . . . . . . . . . 22 (π‘Ž ∈ Word 𝑉 β†’ (lastSβ€˜π‘Ž) = (π‘Žβ€˜((β™―β€˜π‘Ž) βˆ’ 1)))
8281adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) β†’ (lastSβ€˜π‘Ž) = (π‘Žβ€˜((β™―β€˜π‘Ž) βˆ’ 1)))
83 oveq1 7369 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (β™―β€˜π‘Ž) β†’ (𝑁 βˆ’ 1) = ((β™―β€˜π‘Ž) βˆ’ 1))
8483eqcoms 2745 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜π‘Ž) = 𝑁 β†’ (𝑁 βˆ’ 1) = ((β™―β€˜π‘Ž) βˆ’ 1))
8584fveq2d 6851 . . . . . . . . . . . . . . . . . . . . . . 23 ((β™―β€˜π‘Ž) = 𝑁 β†’ (π‘Žβ€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜((β™―β€˜π‘Ž) βˆ’ 1)))
8685eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜π‘Ž) = 𝑁 β†’ ((lastSβ€˜π‘Ž) = (π‘Žβ€˜(𝑁 βˆ’ 1)) ↔ (lastSβ€˜π‘Ž) = (π‘Žβ€˜((β™―β€˜π‘Ž) βˆ’ 1))))
8786adantl 483 . . . . . . . . . . . . . . . . . . . . 21 ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) β†’ ((lastSβ€˜π‘Ž) = (π‘Žβ€˜(𝑁 βˆ’ 1)) ↔ (lastSβ€˜π‘Ž) = (π‘Žβ€˜((β™―β€˜π‘Ž) βˆ’ 1))))
8882, 87mpbird 257 . . . . . . . . . . . . . . . . . . . 20 ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) β†’ (lastSβ€˜π‘Ž) = (π‘Žβ€˜(𝑁 βˆ’ 1)))
8936, 88syl 17 . . . . . . . . . . . . . . . . . . 19 (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) β†’ (lastSβ€˜π‘Ž) = (π‘Žβ€˜(𝑁 βˆ’ 1)))
9089eqcomd 2743 . . . . . . . . . . . . . . . . . 18 (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) β†’ (π‘Žβ€˜(𝑁 βˆ’ 1)) = (lastSβ€˜π‘Ž))
9190adantr 482 . . . . . . . . . . . . . . . . 17 ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) β†’ (π‘Žβ€˜(𝑁 βˆ’ 1)) = (lastSβ€˜π‘Ž))
9291ad2antrl 727 . . . . . . . . . . . . . . . 16 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (π‘Žβ€˜(𝑁 βˆ’ 1)) = (lastSβ€˜π‘Ž))
9380, 92eqeq12d 2753 . . . . . . . . . . . . . . 15 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ ((π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)) ↔ (lastSβ€˜π‘) = (lastSβ€˜π‘Ž)))
9493biimpd 228 . . . . . . . . . . . . . 14 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ ((π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)) β†’ (lastSβ€˜π‘) = (lastSβ€˜π‘Ž)))
9594adantld 492 . . . . . . . . . . . . 13 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1))) β†’ (lastSβ€˜π‘) = (lastSβ€˜π‘Ž)))
9695imp 408 . . . . . . . . . . . 12 (((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ (lastSβ€˜π‘) = (lastSβ€˜π‘Ž))
9764, 74, 963jca 1129 . . . . . . . . . . 11 (((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ ((𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2)) ∧ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = (π‘Žβ€˜((β™―β€˜π‘) βˆ’ 2)) ∧ (lastSβ€˜π‘) = (lastSβ€˜π‘Ž)))
98973adant1 1131 . . . . . . . . . 10 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ ((𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2)) ∧ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = (π‘Žβ€˜((β™―β€˜π‘) βˆ’ 2)) ∧ (lastSβ€˜π‘) = (lastSβ€˜π‘Ž)))
991clwwlknwrd 29020 . . . . . . . . . . . . 13 (𝑝 ∈ (𝑁 ClWWalksN 𝐺) β†’ 𝑝 ∈ Word 𝑉)
10099ad3antrrr 729 . . . . . . . . . . . 12 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ 𝑝 ∈ Word 𝑉)
1011003ad2ant2 1135 . . . . . . . . . . 11 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ 𝑝 ∈ Word 𝑉)
1021clwwlknwrd 29020 . . . . . . . . . . . . . 14 (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) β†’ π‘Ž ∈ Word 𝑉)
103102adantr 482 . . . . . . . . . . . . 13 ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) β†’ π‘Ž ∈ Word 𝑉)
104103ad2antrl 727 . . . . . . . . . . . 12 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ π‘Ž ∈ Word 𝑉)
1051043ad2ant2 1135 . . . . . . . . . . 11 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ π‘Ž ∈ Word 𝑉)
106 clwwlknlen 29018 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝑁 ClWWalksN 𝐺) β†’ (β™―β€˜π‘) = 𝑁)
107 eluz2b1 12851 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (β„€β‰₯β€˜2) ↔ (𝑁 ∈ β„€ ∧ 1 < 𝑁))
108 breq2 5114 . . . . . . . . . . . . . . . . . 18 (𝑁 = (β™―β€˜π‘) β†’ (1 < 𝑁 ↔ 1 < (β™―β€˜π‘)))
109108eqcoms 2745 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘) = 𝑁 β†’ (1 < 𝑁 ↔ 1 < (β™―β€˜π‘)))
110109biimpcd 249 . . . . . . . . . . . . . . . 16 (1 < 𝑁 β†’ ((β™―β€˜π‘) = 𝑁 β†’ 1 < (β™―β€˜π‘)))
111107, 110simplbiim 506 . . . . . . . . . . . . . . 15 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ ((β™―β€˜π‘) = 𝑁 β†’ 1 < (β™―β€˜π‘)))
11214, 106, 111syl2imc 41 . . . . . . . . . . . . . 14 (𝑝 ∈ (𝑁 ClWWalksN 𝐺) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 1 < (β™―β€˜π‘)))
113112ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 1 < (β™―β€˜π‘)))
114113impcom 409 . . . . . . . . . . . 12 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋))) β†’ 1 < (β™―β€˜π‘))
1151143adant3 1133 . . . . . . . . . . 11 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ 1 < (β™―β€˜π‘))
116 2swrd2eqwrdeq 14849 . . . . . . . . . . 11 ((𝑝 ∈ Word 𝑉 ∧ π‘Ž ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘)) β†’ (𝑝 = π‘Ž ↔ ((β™―β€˜π‘) = (β™―β€˜π‘Ž) ∧ ((𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2)) ∧ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = (π‘Žβ€˜((β™―β€˜π‘) βˆ’ 2)) ∧ (lastSβ€˜π‘) = (lastSβ€˜π‘Ž)))))
117101, 105, 115, 116syl3anc 1372 . . . . . . . . . 10 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ (𝑝 = π‘Ž ↔ ((β™―β€˜π‘) = (β™―β€˜π‘Ž) ∧ ((𝑝 prefix ((β™―β€˜π‘) βˆ’ 2)) = (π‘Ž prefix ((β™―β€˜π‘) βˆ’ 2)) ∧ (π‘β€˜((β™―β€˜π‘) βˆ’ 2)) = (π‘Žβ€˜((β™―β€˜π‘) βˆ’ 2)) ∧ (lastSβ€˜π‘) = (lastSβ€˜π‘Ž)))))
11853, 98, 117mpbir2and 712 . . . . . . . . 9 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) ∧ ((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1)))) β†’ 𝑝 = π‘Ž)
1191183exp 1120 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1))) β†’ 𝑝 = π‘Ž)))
1201193ad2ant3 1136 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) ∧ ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)) β†’ (((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1))) β†’ 𝑝 = π‘Ž)))
12125, 120sylbid 239 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁)) β†’ (((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1))) β†’ 𝑝 = π‘Ž)))
122121imp 408 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁))) β†’ (((𝑝 prefix (𝑁 βˆ’ 2)) = (π‘Ž prefix (𝑁 βˆ’ 2)) ∧ (π‘β€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜(𝑁 βˆ’ 1))) β†’ 𝑝 = π‘Ž))
12313, 122biimtrid 241 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁))) β†’ (⟨(𝑝 prefix (𝑁 βˆ’ 2)), (π‘β€˜(𝑁 βˆ’ 1))⟩ = ⟨(π‘Ž prefix (𝑁 βˆ’ 2)), (π‘Žβ€˜(𝑁 βˆ’ 1))⟩ β†’ 𝑝 = π‘Ž))
12410, 123sylbid 239 . . 3 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (𝑝 ∈ (𝑋𝐢𝑁) ∧ π‘Ž ∈ (𝑋𝐢𝑁))) β†’ ((π‘‡β€˜π‘) = (π‘‡β€˜π‘Ž) β†’ 𝑝 = π‘Ž))
125124ralrimivva 3198 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βˆ€π‘ ∈ (𝑋𝐢𝑁)βˆ€π‘Ž ∈ (𝑋𝐢𝑁)((π‘‡β€˜π‘) = (π‘‡β€˜π‘Ž) β†’ 𝑝 = π‘Ž))
126 dff13 7207 . 2 (𝑇:(𝑋𝐢𝑁)–1-1β†’(𝐹 Γ— (𝐺 NeighbVtx 𝑋)) ↔ (𝑇:(𝑋𝐢𝑁)⟢(𝐹 Γ— (𝐺 NeighbVtx 𝑋)) ∧ βˆ€π‘ ∈ (𝑋𝐢𝑁)βˆ€π‘Ž ∈ (𝑋𝐢𝑁)((π‘‡β€˜π‘) = (π‘‡β€˜π‘Ž) β†’ 𝑝 = π‘Ž)))
1275, 125, 126sylanbrc 584 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)–1-1β†’(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  βŸ¨cop 4597   class class class wbr 5110   ↦ cmpt 5193   Γ— cxp 5636  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  0cc0 11058  1c1 11059   < clt 11196   βˆ’ cmin 11392  2c2 12215  3c3 12216  β„€cz 12506  β„€β‰₯cuz 12770  β™―chash 14237  Word cword 14409  lastSclsw 14457   prefix cpfx 14565  Vtxcvtx 27989  USGraphcusgr 28142   NeighbVtx cnbgr 28322   ClWWalksN cclwwlkn 29010  ClWWalksNOncclwwlknon 29073
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-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
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-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-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-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-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-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  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-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-rp 12923  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-lsw 14458  df-concat 14466  df-s1 14491  df-substr 14536  df-pfx 14566  df-s2 14744  df-edg 28041  df-upgr 28075  df-umgr 28076  df-usgr 28144  df-nbgr 28323  df-wwlks 28817  df-wwlksn 28818  df-clwwlk 28968  df-clwwlkn 29011  df-clwwlknon 29074
This theorem is referenced by:  numclwwlk1lem2f1o  29345
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