MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numclwwlk1lem2f1 Structured version   Visualization version   GIF version

Theorem numclwwlk1lem2f1 30154
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 30152 . 2 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)⟢(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
61, 2, 3, 4numclwwlk1lem2fv 30153 . . . . . 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 30153 . . . . . 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 2743 . . . 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 7447 . . . . . 6 (𝑝 prefix (𝑁 βˆ’ 2)) ∈ V
12 fvex 6904 . . . . . 6 (π‘β€˜(𝑁 βˆ’ 1)) ∈ V
1311, 12opth 5472 . . . . 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 12895 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
1522clwwlkel 30146 . . . . . . . . . . 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 29888 . . . . . . . . . . . 12 (𝑝 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ (𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋))
1716anbi1i 623 . . . . . . . . . . 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 30146 . . . . . . . . . . 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 29888 . . . . . . . . . . . 12 (π‘Ž ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋))
2120anbi1i 623 . . . . . . . . . . 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 630 . . . . . . . . 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 592 . . . . . . . 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 1128 . . . . . . 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 29832 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝑁 ClWWalksN 𝐺) β†’ (𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁))
2726adantr 480 . . . . . . . . . . . . . 14 ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) β†’ (𝑝 ∈ Word 𝑉 ∧ (β™―β€˜π‘) = 𝑁))
2827adantr 480 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) β†’ (π‘β€˜0) = 𝑋)
3029adantr 480 . . . . . . . . . . . . 13 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘β€˜0) = 𝑋)
31 simpr 484 . . . . . . . . . . . . . 14 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋)
3229eqcomd 2733 . . . . . . . . . . . . . . 15 ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) β†’ 𝑋 = (π‘β€˜0))
3332adantr 480 . . . . . . . . . . . . . 14 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑋 = (π‘β€˜0))
3431, 33eqtrd 2767 . . . . . . . . . . . . 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 515 . . . . . . . . . . . 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 29832 . . . . . . . . . . . . . . 15 (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) β†’ (π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁))
3736adantr 480 . . . . . . . . . . . . . 14 ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) β†’ (π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁))
3837adantr 480 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) β†’ (π‘Žβ€˜0) = 𝑋)
4039adantr 480 . . . . . . . . . . . . 13 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘Žβ€˜0) = 𝑋)
41 simpr 484 . . . . . . . . . . . . . 14 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋)
4239eqcomd 2733 . . . . . . . . . . . . . . 15 ((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) β†’ 𝑋 = (π‘Žβ€˜0))
4342adantr 480 . . . . . . . . . . . . . 14 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑋 = (π‘Žβ€˜0))
4441, 43eqtrd 2767 . . . . . . . . . . . . 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 515 . . . . . . . . . . . 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 2753 . . . . . . . . . . . . . . . . 17 (((β™―β€˜π‘) = 𝑁 ∧ (β™―β€˜π‘Ž) = 𝑁) β†’ (β™―β€˜π‘) = (β™―β€˜π‘Ž))
4746expcom 413 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . 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 595 . . . . . . . . . . 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 1132 . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) β†’ (β™―β€˜π‘) = 𝑁)
5554adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ (β™―β€˜π‘) = 𝑁)
5655eqcomd 2733 . . . . . . . . . . . . . . . . . . 19 (((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘β€˜0) = 𝑋) ∧ (π‘β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑁 = (β™―β€˜π‘))
5756adantr 480 . . . . . . . . . . . . . . . . . 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 7429 . . . . . . . . . . . . . . . . 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 7430 . . . . . . . . . . . . . . . 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 7430 . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 407 . . . . . . . . . . . 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 7429 . . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . 16 (((π‘Ž ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘Žβ€˜0) = 𝑋) ∧ (π‘Žβ€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ 𝑋 = (π‘Žβ€˜(𝑁 βˆ’ 2)))
7069adantl 481 . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 14538 . . . . . . . . . . . . . . . . . . . 20 (𝑝 ∈ Word 𝑉 β†’ (lastSβ€˜π‘) = (π‘β€˜((β™―β€˜π‘) βˆ’ 1)))
76 fvoveq1 7437 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜π‘) = 𝑁 β†’ (π‘β€˜((β™―β€˜π‘) βˆ’ 1)) = (π‘β€˜(𝑁 βˆ’ 1)))
7775, 76sylan9eq 2787 . . . . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . 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 14538 . . . . . . . . . . . . . . . . . . . . . 22 (π‘Ž ∈ Word 𝑉 β†’ (lastSβ€˜π‘Ž) = (π‘Žβ€˜((β™―β€˜π‘Ž) βˆ’ 1)))
8281adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((π‘Ž ∈ Word 𝑉 ∧ (β™―β€˜π‘Ž) = 𝑁) β†’ (lastSβ€˜π‘Ž) = (π‘Žβ€˜((β™―β€˜π‘Ž) βˆ’ 1)))
83 oveq1 7421 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (β™―β€˜π‘Ž) β†’ (𝑁 βˆ’ 1) = ((β™―β€˜π‘Ž) βˆ’ 1))
8483eqcoms 2735 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β™―β€˜π‘Ž) = 𝑁 β†’ (𝑁 βˆ’ 1) = ((β™―β€˜π‘Ž) βˆ’ 1))
8584fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . . 23 ((β™―β€˜π‘Ž) = 𝑁 β†’ (π‘Žβ€˜(𝑁 βˆ’ 1)) = (π‘Žβ€˜((β™―β€˜π‘Ž) βˆ’ 1)))
8685eqeq2d 2738 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . . 18 (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) β†’ (π‘Žβ€˜(𝑁 βˆ’ 1)) = (lastSβ€˜π‘Ž))
9190adantr 480 . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . 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 406 . . . . . . . . . . . 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 1126 . . . . . . . . . . 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 1128 . . . . . . . . . 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 29831 . . . . . . . . . . . . 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 1132 . . . . . . . . . . 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 29831 . . . . . . . . . . . . . 14 (π‘Ž ∈ (𝑁 ClWWalksN 𝐺) β†’ π‘Ž ∈ Word 𝑉)
103102adantr 480 . . . . . . . . . . . . 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 1132 . . . . . . . . . . 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 29829 . . . . . . . . . . . . . . 15 (𝑝 ∈ (𝑁 ClWWalksN 𝐺) β†’ (β™―β€˜π‘) = 𝑁)
107 eluz2b1 12925 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (β„€β‰₯β€˜2) ↔ (𝑁 ∈ β„€ ∧ 1 < 𝑁))
108 breq2 5146 . . . . . . . . . . . . . . . . . 18 (𝑁 = (β™―β€˜π‘) β†’ (1 < 𝑁 ↔ 1 < (β™―β€˜π‘)))
109108eqcoms 2735 . . . . . . . . . . . . . . . . 17 ((β™―β€˜π‘) = 𝑁 β†’ (1 < 𝑁 ↔ 1 < (β™―β€˜π‘)))
110109biimpcd 248 . . . . . . . . . . . . . . . 16 (1 < 𝑁 β†’ ((β™―β€˜π‘) = 𝑁 β†’ 1 < (β™―β€˜π‘)))
111107, 110simplbiim 504 . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . 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 1130 . . . . . . . . . . 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 14928 . . . . . . . . . . 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 1369 . . . . . . . . . 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 1117 . . . . . . . 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 1133 . . . . . . 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 406 . . . . 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 3195 . 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 7259 . 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 582 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  {crab 3427  βŸ¨cop 4630   class class class wbr 5142   ↦ cmpt 5225   Γ— cxp 5670  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  0cc0 11130  1c1 11131   < clt 11270   βˆ’ cmin 11466  2c2 12289  3c3 12290  β„€cz 12580  β„€β‰₯cuz 12844  β™―chash 14313  Word cword 14488  lastSclsw 14536   prefix cpfx 14644  Vtxcvtx 28796  USGraphcusgr 28949   NeighbVtx cnbgr 29132   ClWWalksN cclwwlkn 29821  ClWWalksNOncclwwlknon 29884
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-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
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-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-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-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-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-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  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-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-s1 14570  df-substr 14615  df-pfx 14645  df-s2 14823  df-edg 28848  df-upgr 28882  df-umgr 28883  df-usgr 28951  df-nbgr 29133  df-wwlks 29628  df-wwlksn 29629  df-clwwlk 29779  df-clwwlkn 29822  df-clwwlknon 29885
This theorem is referenced by:  numclwwlk1lem2f1o  30156
  Copyright terms: Public domain W3C validator