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Theorem numclwwlk1lem2fo 29344
Description: 𝑇 is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 13-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
numclwwlk1lem2fo ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)–ontoβ†’(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 numclwwlk1lem2fo
Dummy variables 𝑖 π‘Ž 𝑝 𝑏 π‘₯ 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)))
6 elxp 5661 . . . . 5 (𝑝 ∈ (𝐹 Γ— (𝐺 NeighbVtx 𝑋)) ↔ βˆƒπ‘Žβˆƒπ‘(𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))))
71, 2, 3numclwwlk1lem2foa 29340 . . . . . . . . . . 11 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋)) β†’ ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁)))
87com12 32 . . . . . . . . . 10 ((π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋)) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁)))
98adantl 483 . . . . . . . . 9 ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁)))
109imp 408 . . . . . . . 8 (((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁))
11 simpl 484 . . . . . . . . 9 ((((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) ∧ ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))) β†’ ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁))
12 fveq2 6847 . . . . . . . . . . 11 (π‘₯ = ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) β†’ (π‘‡β€˜π‘₯) = (π‘‡β€˜((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)))
1312eqeq2d 2748 . . . . . . . . . 10 (π‘₯ = ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) β†’ (𝑝 = (π‘‡β€˜π‘₯) ↔ 𝑝 = (π‘‡β€˜((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©))))
141, 2, 3, 4numclwwlk1lem2fv 29342 . . . . . . . . . . . 12 (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) β†’ (π‘‡β€˜((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)) = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)
1514adantr 482 . . . . . . . . . . 11 ((((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) ∧ ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))) β†’ (π‘‡β€˜((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)) = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)
1615eqeq2d 2748 . . . . . . . . . 10 ((((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) ∧ ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))) β†’ (𝑝 = (π‘‡β€˜((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)) ↔ 𝑝 = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩))
1713, 16sylan9bbr 512 . . . . . . . . 9 (((((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) ∧ ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))) ∧ π‘₯ = ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)) β†’ (𝑝 = (π‘‡β€˜π‘₯) ↔ 𝑝 = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩))
18 simprll 778 . . . . . . . . . 10 ((((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) ∧ ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))) β†’ 𝑝 = βŸ¨π‘Ž, π‘βŸ©)
191nbgrisvtx 28331 . . . . . . . . . . . . . . . 16 (𝑏 ∈ (𝐺 NeighbVtx 𝑋) β†’ 𝑏 ∈ 𝑉)
203eleq2i 2830 . . . . . . . . . . . . . . . . . . . 20 (π‘Ž ∈ 𝐹 ↔ π‘Ž ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))
21 uz3m2nn 12823 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 2) ∈ β„•)
2221nnne0d 12210 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 2) β‰  0)
23223ad2ant3 1136 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑁 βˆ’ 2) β‰  0)
24 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
251, 24clwwlknonel 29081 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 βˆ’ 2) β‰  0 β†’ (π‘Ž ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ↔ ((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2) ∧ (π‘Žβ€˜0) = 𝑋)))
2623, 25syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Ž ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ↔ ((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2) ∧ (π‘Žβ€˜0) = 𝑋)))
2720, 26bitrid 283 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Ž ∈ 𝐹 ↔ ((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2) ∧ (π‘Žβ€˜0) = 𝑋)))
28 df-3an 1090 . . . . . . . . . . . . . . . . . . 19 (((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2) ∧ (π‘Žβ€˜0) = 𝑋) ↔ (((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (π‘Žβ€˜0) = 𝑋))
2927, 28bitrdi 287 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Ž ∈ 𝐹 ↔ (((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (π‘Žβ€˜0) = 𝑋)))
30 simplll 774 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ π‘Ž ∈ Word 𝑉)
31 s1cl 14497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑋 ∈ 𝑉 β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉)
3231adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉)
3332adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉)
3433adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉)
35 s1cl 14497 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑏 ∈ 𝑉 β†’ βŸ¨β€œπ‘β€βŸ© ∈ Word 𝑉)
3635adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ βŸ¨β€œπ‘β€βŸ© ∈ Word 𝑉)
37 ccatass 14483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((π‘Ž ∈ Word 𝑉 ∧ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉 ∧ βŸ¨β€œπ‘β€βŸ© ∈ Word 𝑉) β†’ ((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) = (π‘Ž ++ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘β€βŸ©)))
3837oveq1d 7377 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘Ž ∈ Word 𝑉 ∧ βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉 ∧ βŸ¨β€œπ‘β€βŸ© ∈ Word 𝑉) β†’ (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)) = ((π‘Ž ++ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘β€βŸ©)) prefix (𝑁 βˆ’ 2)))
3930, 34, 36, 38syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)) = ((π‘Ž ++ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘β€βŸ©)) prefix (𝑁 βˆ’ 2)))
40 ccatcl 14469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((βŸ¨β€œπ‘‹β€βŸ© ∈ Word 𝑉 ∧ βŸ¨β€œπ‘β€βŸ© ∈ Word 𝑉) β†’ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘β€βŸ©) ∈ Word 𝑉)
4133, 35, 40syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘β€βŸ©) ∈ Word 𝑉)
42 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) β†’ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2))
4342eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) β†’ (𝑁 βˆ’ 2) = (β™―β€˜π‘Ž))
4443adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝑁 βˆ’ 2) = (β™―β€˜π‘Ž))
4544adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ (𝑁 βˆ’ 2) = (β™―β€˜π‘Ž))
46 pfxccatid 14636 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((π‘Ž ∈ Word 𝑉 ∧ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘β€βŸ©) ∈ Word 𝑉 ∧ (𝑁 βˆ’ 2) = (β™―β€˜π‘Ž)) β†’ ((π‘Ž ++ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘β€βŸ©)) prefix (𝑁 βˆ’ 2)) = π‘Ž)
4730, 41, 45, 46syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ ((π‘Ž ++ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘β€βŸ©)) prefix (𝑁 βˆ’ 2)) = π‘Ž)
4839, 47eqtr2d 2778 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ π‘Ž = (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)))
49 1e2m1 12287 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 = (2 βˆ’ 1)
5049a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 1 = (2 βˆ’ 1))
5150oveq2d 7378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 1) = (𝑁 βˆ’ (2 βˆ’ 1)))
52 eluzelcn 12782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ β„‚)
53 2cnd 12238 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 2 ∈ β„‚)
54 1cnd 11157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 1 ∈ β„‚)
5552, 53, 54subsubd 11547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ (2 βˆ’ 1)) = ((𝑁 βˆ’ 2) + 1))
5651, 55eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 1) = ((𝑁 βˆ’ 2) + 1))
5756adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑁 βˆ’ 1) = ((𝑁 βˆ’ 2) + 1))
5857adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝑁 βˆ’ 1) = ((𝑁 βˆ’ 2) + 1))
5958adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ (𝑁 βˆ’ 1) = ((𝑁 βˆ’ 2) + 1))
6059fveq2d 6851 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1)) = (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜((𝑁 βˆ’ 2) + 1)))
61 simpll 766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ (π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)))
62 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝑋 ∈ 𝑉)
6362anim1i 616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ (𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
64 ccatw2s1p2 14532 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) β†’ (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜((𝑁 βˆ’ 2) + 1)) = 𝑏)
6561, 63, 64syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜((𝑁 βˆ’ 2) + 1)) = 𝑏)
6660, 65eqtr2d 2778 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ 𝑏 = (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1)))
6748, 66opeq12d 4843 . . . . . . . . . . . . . . . . . . . . . . 23 ((((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ 𝑏 ∈ 𝑉) β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)
6867exp31 421 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) β†’ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑏 ∈ 𝑉 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
69683ad2antl1 1186 . . . . . . . . . . . . . . . . . . . . 21 (((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) β†’ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑏 ∈ 𝑉 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
7069adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (π‘Žβ€˜0) = 𝑋) β†’ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑏 ∈ 𝑉 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
7170com12 32 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (π‘Žβ€˜0) = 𝑋) β†’ (𝑏 ∈ 𝑉 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
72713adant1 1131 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((((π‘Ž ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Ž) βˆ’ 1)){(π‘Žβ€˜π‘–), (π‘Žβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Ž), (π‘Žβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Ž) = (𝑁 βˆ’ 2)) ∧ (π‘Žβ€˜0) = 𝑋) β†’ (𝑏 ∈ 𝑉 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
7329, 72sylbid 239 . . . . . . . . . . . . . . . . 17 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Ž ∈ 𝐹 β†’ (𝑏 ∈ 𝑉 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
7473com23 86 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑏 ∈ 𝑉 β†’ (π‘Ž ∈ 𝐹 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
7519, 74syl5 34 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑏 ∈ (𝐺 NeighbVtx 𝑋) β†’ (π‘Ž ∈ 𝐹 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
7675com13 88 . . . . . . . . . . . . . 14 (π‘Ž ∈ 𝐹 β†’ (𝑏 ∈ (𝐺 NeighbVtx 𝑋) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)))
7776imp 408 . . . . . . . . . . . . 13 ((π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋)) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩))
7877adantl 483 . . . . . . . . . . . 12 ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩))
7978imp 408 . . . . . . . . . . 11 (((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)
8079adantl 483 . . . . . . . . . 10 ((((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) ∧ ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))) β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)
8118, 80eqtrd 2777 . . . . . . . . 9 ((((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) ∧ ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))) β†’ 𝑝 = ⟨(((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) prefix (𝑁 βˆ’ 2)), (((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©)β€˜(𝑁 βˆ’ 1))⟩)
8211, 17, 81rspcedvd 3586 . . . . . . . 8 ((((π‘Ž ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘β€βŸ©) ∈ (𝑋𝐢𝑁) ∧ ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))) β†’ βˆƒπ‘₯ ∈ (𝑋𝐢𝑁)𝑝 = (π‘‡β€˜π‘₯))
8310, 82mpancom 687 . . . . . . 7 (((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ βˆƒπ‘₯ ∈ (𝑋𝐢𝑁)𝑝 = (π‘‡β€˜π‘₯))
8483ex 414 . . . . . 6 ((𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βˆƒπ‘₯ ∈ (𝑋𝐢𝑁)𝑝 = (π‘‡β€˜π‘₯)))
8584exlimivv 1936 . . . . 5 (βˆƒπ‘Žβˆƒπ‘(𝑝 = βŸ¨π‘Ž, π‘βŸ© ∧ (π‘Ž ∈ 𝐹 ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βˆƒπ‘₯ ∈ (𝑋𝐢𝑁)𝑝 = (π‘‡β€˜π‘₯)))
866, 85sylbi 216 . . . 4 (𝑝 ∈ (𝐹 Γ— (𝐺 NeighbVtx 𝑋)) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βˆƒπ‘₯ ∈ (𝑋𝐢𝑁)𝑝 = (π‘‡β€˜π‘₯)))
8786impcom 409 . . 3 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ 𝑝 ∈ (𝐹 Γ— (𝐺 NeighbVtx 𝑋))) β†’ βˆƒπ‘₯ ∈ (𝑋𝐢𝑁)𝑝 = (π‘‡β€˜π‘₯))
8887ralrimiva 3144 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ βˆ€π‘ ∈ (𝐹 Γ— (𝐺 NeighbVtx 𝑋))βˆƒπ‘₯ ∈ (𝑋𝐢𝑁)𝑝 = (π‘‡β€˜π‘₯))
89 dffo3 7057 . 2 (𝑇:(𝑋𝐢𝑁)–ontoβ†’(𝐹 Γ— (𝐺 NeighbVtx 𝑋)) ↔ (𝑇:(𝑋𝐢𝑁)⟢(𝐹 Γ— (𝐺 NeighbVtx 𝑋)) ∧ βˆ€π‘ ∈ (𝐹 Γ— (𝐺 NeighbVtx 𝑋))βˆƒπ‘₯ ∈ (𝑋𝐢𝑁)𝑝 = (π‘‡β€˜π‘₯)))
905, 88, 89sylanbrc 584 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)–ontoβ†’(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  {cpr 4593  βŸ¨cop 4597   ↦ cmpt 5193   Γ— cxp 5636  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  0cc0 11058  1c1 11059   + caddc 11061   βˆ’ cmin 11392  2c2 12215  3c3 12216  β„€β‰₯cuz 12770  ..^cfzo 13574  β™―chash 14237  Word cword 14409  lastSclsw 14457   ++ cconcat 14465  βŸ¨β€œcs1 14490   prefix cpfx 14565  Vtxcvtx 27989  Edgcedg 28040  USGraphcusgr 28142   NeighbVtx cnbgr 28322  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-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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