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Theorem numclwwlk1lem2foa 30151
Description: Going forth and back from the end of a (closed) walk: π‘Š represents the closed walk p0, ..., p(n-2), p0 = p(n-2). With 𝑋 = p(n-2) = p0 and π‘Œ = p(n-1), ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) represents the closed walk p0, ..., p(n-2), p(n-1), pn = p0 which is a double loop of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 5-Mar-2022.) (Proof shortened by AV, 2-Nov-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))
Assertion
Ref Expression
numclwwlk1lem2foa ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑋𝐢𝑁)))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀   𝑛,𝑋,𝑣,𝑀   𝑀,𝐹   𝑀,π‘Š   𝑀,π‘Œ
Allowed substitution hints:   𝐢(𝑀,𝑣,𝑛)   𝐹(𝑣,𝑛)   π‘Š(𝑣,𝑛)   π‘Œ(𝑣,𝑛)

Proof of Theorem numclwwlk1lem2foa
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpl2 1190 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ 𝑋 ∈ 𝑉)
2 extwwlkfab.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
32nbgrisvtx 29141 . . . . 5 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ π‘Œ ∈ 𝑉)
43ad2antll 728 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ π‘Œ ∈ 𝑉)
5 simpl3 1191 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ 𝑁 ∈ (β„€β‰₯β€˜3))
6 nbgrsym 29163 . . . . . . . 8 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx π‘Œ))
7 eqid 2727 . . . . . . . . . 10 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
87nbusgreledg 29153 . . . . . . . . 9 (𝐺 ∈ USGraph β†’ (𝑋 ∈ (𝐺 NeighbVtx π‘Œ) ↔ {𝑋, π‘Œ} ∈ (Edgβ€˜πΊ)))
98biimpd 228 . . . . . . . 8 (𝐺 ∈ USGraph β†’ (𝑋 ∈ (𝐺 NeighbVtx π‘Œ) β†’ {𝑋, π‘Œ} ∈ (Edgβ€˜πΊ)))
106, 9biimtrid 241 . . . . . . 7 (𝐺 ∈ USGraph β†’ (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ {𝑋, π‘Œ} ∈ (Edgβ€˜πΊ)))
1110adantld 490 . . . . . 6 (𝐺 ∈ USGraph β†’ ((π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ {𝑋, π‘Œ} ∈ (Edgβ€˜πΊ)))
12113ad2ant1 1131 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ {𝑋, π‘Œ} ∈ (Edgβ€˜πΊ)))
1312imp 406 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ {𝑋, π‘Œ} ∈ (Edgβ€˜πΊ))
14 simprl 770 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ π‘Š ∈ 𝐹)
15 extwwlkfab.f . . . . 5 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))
1614, 15eleqtrdi 2838 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))
172, 7clwwlknonex2 29906 . . . 4 (((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ {𝑋, π‘Œ} ∈ (Edgβ€˜πΊ) ∧ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑁 ClWWalksN 𝐺))
181, 4, 5, 13, 16, 17syl311anc 1382 . . 3 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑁 ClWWalksN 𝐺))
1915eleq2i 2820 . . . . . . . 8 (π‘Š ∈ 𝐹 ↔ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))
20 uz3m2nn 12897 . . . . . . . . . . 11 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 2) ∈ β„•)
2120nnne0d 12284 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 2) β‰  0)
222, 7clwwlknonel 29892 . . . . . . . . . 10 ((𝑁 βˆ’ 2) β‰  0 β†’ (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ↔ ((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (π‘Šβ€˜0) = 𝑋)))
2321, 22syl 17 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ↔ ((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (π‘Šβ€˜0) = 𝑋)))
24233ad2ant3 1133 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ↔ ((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (π‘Šβ€˜0) = 𝑋)))
2519, 24bitrid 283 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Š ∈ 𝐹 ↔ ((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (π‘Šβ€˜0) = 𝑋)))
26 3simpa 1146 . . . . . . . . . . . . . 14 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)))
2726adantr 480 . . . . . . . . . . . . 13 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)))
28 simp32 1208 . . . . . . . . . . . . . 14 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝑋 ∈ 𝑉)
2928, 3anim12i 612 . . . . . . . . . . . . 13 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉))
30 simpl33 1254 . . . . . . . . . . . . 13 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ 𝑁 ∈ (β„€β‰₯β€˜3))
3127, 29, 303jca 1126 . . . . . . . . . . . 12 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))
32313exp1 1350 . . . . . . . . . . 11 (π‘Š ∈ Word 𝑉 β†’ ((β™―β€˜π‘Š) = (𝑁 βˆ’ 2) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3))))))
33323ad2ant1 1131 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) β†’ ((β™―β€˜π‘Š) = (𝑁 βˆ’ 2) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3))))))
3433imp 406 . . . . . . . . 9 (((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))))
35343adant3 1130 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (π‘Šβ€˜0) = 𝑋) β†’ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))))
3635com12 32 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (((π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2) ∧ (π‘Šβ€˜0) = 𝑋) β†’ (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))))
3725, 36sylbid 239 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Š ∈ 𝐹 β†’ (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))))
3837imp32 418 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)))
39 numclwwlk1lem2foalem 30148 . . . . 5 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) = π‘Š ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) = π‘Œ ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋))
4038, 39syl 17 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) = π‘Š ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) = π‘Œ ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋))
41 eleq1a 2823 . . . . . 6 (π‘Š ∈ 𝐹 β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) = π‘Š β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) ∈ 𝐹))
4214, 41syl 17 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) = π‘Š β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) ∈ 𝐹))
43 eleq1a 2823 . . . . . 6 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) = π‘Œ β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋)))
4443ad2antll 728 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) = π‘Œ β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋)))
45 idd 24 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋 β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋))
4642, 44, 453anim123d 1440 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ (((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) = π‘Š ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) = π‘Œ ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋)))
4740, 46mpd 15 . . 3 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋))
48 extwwlkfab.c . . . . 5 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
492, 48, 15extwwlkfabel 30150 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑋𝐢𝑁) ↔ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑁 ClWWalksN 𝐺) ∧ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋))))
5049adantr 480 . . 3 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑋𝐢𝑁) ↔ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑁 ClWWalksN 𝐺) ∧ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋))))
5118, 47, 50mpbir2and 712 . 2 (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) ∧ (π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋))) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑋𝐢𝑁))
5251ex 412 1 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑋𝐢𝑁)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆ€wral 3056  {crab 3427  {cpr 4626  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  0cc0 11130  1c1 11131   + caddc 11133   βˆ’ cmin 11466  2c2 12289  3c3 12290  β„€β‰₯cuz 12844  ..^cfzo 13651  β™―chash 14313  Word cword 14488  lastSclsw 14536   ++ cconcat 14544  βŸ¨β€œcs1 14569   prefix cpfx 14644  Vtxcvtx 28796  Edgcedg 28847  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-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:  numclwwlk1lem2fo  30155
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