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Theorem numclwwlk1lem2foa 29340
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 1193 . . . 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 28331 . . . . 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 1194 . . . 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 28353 . . . . . . . 8 (π‘Œ ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx π‘Œ))
7 eqid 2737 . . . . . . . . . 10 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
87nbusgreledg 28343 . . . . . . . . 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 492 . . . . . 6 (𝐺 ∈ USGraph β†’ ((π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ {𝑋, π‘Œ} ∈ (Edgβ€˜πΊ)))
12113ad2ant1 1134 . . . . 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 408 . . . 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 2848 . . . 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 29095 . . . 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 1385 . . 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 2830 . . . . . . . 8 (π‘Š ∈ 𝐹 ↔ π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))
20 uz3m2nn 12823 . . . . . . . . . . 11 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 2) ∈ β„•)
2120nnne0d 12210 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 2) β‰  0)
222, 7clwwlknonel 29081 . . . . . . . . . 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 1136 . . . . . . . 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 1149 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . . 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 614 . . . . . . . . . . . . 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 1257 . . . . . . . . . . . . 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 1129 . . . . . . . . . . . 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 1353 . . . . . . . . . . 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 1134 . . . . . . . . . 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 408 . . . . . . . . 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 1133 . . . . . . . 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 420 . . . . 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 29337 . . . . 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 2833 . . . . . 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 2833 . . . . . 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 1444 . . . 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 29339 . . . 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 482 . . 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 414 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  {crab 3410  {cpr 4593  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   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-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:  numclwwlk1lem2fo  29344
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