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Theorem numclwwlk2lem1 29897
Description: In a friendship graph, for each walk of length 𝑛 starting at a fixed vertex 𝑣 and ending not at this vertex, there is a unique vertex so that the walk extended by an edge to this vertex and an edge from this vertex to the first vertex of the walk is a value of operation 𝐻. If the walk is represented as a word, it is sufficient to add one vertex to the word to obtain the closed walk contained in the value of operation 𝐻, since in a word representing a closed walk the starting vertex is not repeated at the end. This theorem generally holds only for friendship graphs, because these guarantee that for the first and last vertex there is a (unique) third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 1-May-2022.)
Hypotheses
Ref Expression
numclwwlk.v 𝑉 = (Vtxβ€˜πΊ)
numclwwlk.q 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})
numclwwlk.h 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})
Assertion
Ref Expression
numclwwlk2lem1 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ (𝑋𝑄𝑁) β†’ βˆƒ!𝑣 ∈ 𝑉 (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2))))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑀   𝑀,𝑉   𝑣,π‘Š,𝑀
Allowed substitution hints:   𝑄(𝑀,𝑣,𝑛)   𝐻(𝑀,𝑣,𝑛)   π‘Š(𝑛)

Proof of Theorem numclwwlk2lem1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 numclwwlk.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
2 numclwwlk.q . . . . . 6 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})
31, 2numclwwlkovq 29895 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
433adant1 1129 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
54eleq2d 2818 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ (𝑋𝑄𝑁) ↔ π‘Š ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)}))
6 fveq1 6890 . . . . . 6 (𝑀 = π‘Š β†’ (π‘€β€˜0) = (π‘Šβ€˜0))
76eqeq1d 2733 . . . . 5 (𝑀 = π‘Š β†’ ((π‘€β€˜0) = 𝑋 ↔ (π‘Šβ€˜0) = 𝑋))
8 fveq2 6891 . . . . . 6 (𝑀 = π‘Š β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘Š))
98neeq1d 2999 . . . . 5 (𝑀 = π‘Š β†’ ((lastSβ€˜π‘€) β‰  𝑋 ↔ (lastSβ€˜π‘Š) β‰  𝑋))
107, 9anbi12d 630 . . . 4 (𝑀 = π‘Š β†’ (((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋) ↔ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)))
1110elrab 3683 . . 3 (π‘Š ∈ {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)} ↔ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)))
125, 11bitrdi 287 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ (𝑋𝑄𝑁) ↔ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))))
13 simpl1 1190 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ 𝐺 ∈ FriendGraph )
14 eqid 2731 . . . . . . . . . . . . 13 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
151, 14wwlknp 29365 . . . . . . . . . . . 12 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
16 peano2nn 12229 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„• β†’ (𝑁 + 1) ∈ β„•)
1716adantl 481 . . . . . . . . . . . . . . 15 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ (𝑁 + 1) ∈ β„•)
18 simpl 482 . . . . . . . . . . . . . . 15 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)))
1917, 18jca 511 . . . . . . . . . . . . . 14 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ ((𝑁 + 1) ∈ β„• ∧ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1))))
2019ex 412 . . . . . . . . . . . . 13 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ (𝑁 ∈ β„• β†’ ((𝑁 + 1) ∈ β„• ∧ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)))))
21203adant3 1131 . . . . . . . . . . . 12 ((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) β†’ (𝑁 ∈ β„• β†’ ((𝑁 + 1) ∈ β„• ∧ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)))))
2215, 21syl 17 . . . . . . . . . . 11 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝑁 ∈ β„• β†’ ((𝑁 + 1) ∈ β„• ∧ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)))))
23 lswlgt0cl 14524 . . . . . . . . . . 11 (((𝑁 + 1) ∈ β„• ∧ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1))) β†’ (lastSβ€˜π‘Š) ∈ 𝑉)
2422, 23syl6 35 . . . . . . . . . 10 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝑁 ∈ β„• β†’ (lastSβ€˜π‘Š) ∈ 𝑉))
2524adantr 480 . . . . . . . . 9 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (𝑁 ∈ β„• β†’ (lastSβ€˜π‘Š) ∈ 𝑉))
2625com12 32 . . . . . . . 8 (𝑁 ∈ β„• β†’ ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (lastSβ€˜π‘Š) ∈ 𝑉))
27263ad2ant3 1134 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (lastSβ€˜π‘Š) ∈ 𝑉))
2827imp 406 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ (lastSβ€˜π‘Š) ∈ 𝑉)
29 eleq1 2820 . . . . . . . . . . 11 ((π‘Šβ€˜0) = 𝑋 β†’ ((π‘Šβ€˜0) ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
3029biimprd 247 . . . . . . . . . 10 ((π‘Šβ€˜0) = 𝑋 β†’ (𝑋 ∈ 𝑉 β†’ (π‘Šβ€˜0) ∈ 𝑉))
3130ad2antrl 725 . . . . . . . . 9 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (𝑋 ∈ 𝑉 β†’ (π‘Šβ€˜0) ∈ 𝑉))
3231com12 32 . . . . . . . 8 (𝑋 ∈ 𝑉 β†’ ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (π‘Šβ€˜0) ∈ 𝑉))
33323ad2ant2 1133 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (π‘Šβ€˜0) ∈ 𝑉))
3433imp 406 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ (π‘Šβ€˜0) ∈ 𝑉)
35 neeq2 3003 . . . . . . . . . 10 (𝑋 = (π‘Šβ€˜0) β†’ ((lastSβ€˜π‘Š) β‰  𝑋 ↔ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)))
3635eqcoms 2739 . . . . . . . . 9 ((π‘Šβ€˜0) = 𝑋 β†’ ((lastSβ€˜π‘Š) β‰  𝑋 ↔ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)))
3736biimpa 476 . . . . . . . 8 (((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋) β†’ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))
3837adantl 481 . . . . . . 7 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))
3938adantl 481 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))
4028, 34, 393jca 1127 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ ((lastSβ€˜π‘Š) ∈ 𝑉 ∧ (π‘Šβ€˜0) ∈ 𝑉 ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)))
411, 14frcond2 29788 . . . . 5 (𝐺 ∈ FriendGraph β†’ (((lastSβ€˜π‘Š) ∈ 𝑉 ∧ (π‘Šβ€˜0) ∈ 𝑉 ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)) β†’ βˆƒ!𝑣 ∈ 𝑉 ({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))))
4213, 40, 41sylc 65 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ βˆƒ!𝑣 ∈ 𝑉 ({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
43 simpl 482 . . . . . . . . . 10 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ π‘Š ∈ (𝑁 WWalksN 𝐺))
4443ad2antlr 724 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ π‘Š ∈ (𝑁 WWalksN 𝐺))
45 simpr 484 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ 𝑉)
46 nnnn0 12484 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
47463ad2ant3 1134 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ β„•0)
4847ad2antrr 723 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ 𝑁 ∈ β„•0)
4944, 45, 483jca 1127 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ β„•0))
501, 14wwlksext2clwwlk 29578 . . . . . . . . . 10 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣 ∈ 𝑉) β†’ (({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) β†’ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
51503adant3 1131 . . . . . . . . 9 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) β†’ (({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) β†’ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
5251imp 406 . . . . . . . 8 (((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ β„•0) ∧ ({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))) β†’ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
5349, 52sylan 579 . . . . . . 7 (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ ({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ))) β†’ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
541wwlknbp 29364 . . . . . . . . . . 11 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉))
5554simp3d 1143 . . . . . . . . . 10 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ π‘Š ∈ Word 𝑉)
5655ad2antrl 725 . . . . . . . . 9 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ π‘Š ∈ Word 𝑉)
5756ad2antrr 723 . . . . . . . 8 (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) β†’ π‘Š ∈ Word 𝑉)
5845adantr 480 . . . . . . . 8 (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) β†’ 𝑣 ∈ 𝑉)
59 2z 12599 . . . . . . . . . . 11 2 ∈ β„€
60 nn0pzuz 12894 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ 2 ∈ β„€) β†’ (𝑁 + 2) ∈ (β„€β‰₯β€˜2))
6146, 59, 60sylancl 585 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (𝑁 + 2) ∈ (β„€β‰₯β€˜2))
62613ad2ant3 1134 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑁 + 2) ∈ (β„€β‰₯β€˜2))
6362ad3antrrr 727 . . . . . . . 8 (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) β†’ (𝑁 + 2) ∈ (β„€β‰₯β€˜2))
64 simpr 484 . . . . . . . 8 (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) β†’ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
651, 14clwwlkext2edg 29577 . . . . . . . 8 (((π‘Š ∈ Word 𝑉 ∧ 𝑣 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (β„€β‰₯β€˜2)) ∧ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) β†’ ({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
6657, 58, 63, 64, 65syl31anc 1372 . . . . . . 7 (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) β†’ ({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
6753, 66impbida 798 . . . . . 6 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ↔ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
6845, 1eleqtrdi 2842 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ (Vtxβ€˜πΊ))
6937anim2i 616 . . . . . . . . . . . 12 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)))
7069ad2antlr 724 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)))
7170simprd 495 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))
72 numclwwlk2lem1lem 29863 . . . . . . . . . 10 ((𝑣 ∈ (Vtxβ€˜πΊ) ∧ π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = (π‘Šβ€˜0) ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))
7368, 44, 71, 72syl3anc 1370 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = (π‘Šβ€˜0) ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))
74 eqeq2 2743 . . . . . . . . . . . . 13 (𝑋 = (π‘Šβ€˜0) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = (π‘Šβ€˜0)))
7574eqcoms 2739 . . . . . . . . . . . 12 ((π‘Šβ€˜0) = 𝑋 β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = (π‘Šβ€˜0)))
7675ad2antrl 725 . . . . . . . . . . 11 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = (π‘Šβ€˜0)))
7776ad2antlr 724 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = (π‘Šβ€˜0)))
7873simpld 494 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = (π‘Šβ€˜0))
7978neeq2d 3000 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))
8077, 79anbi12d 630 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ ((((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0)) ↔ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = (π‘Šβ€˜0) ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0))))
8173, 80mpbird 257 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0)))
82 nncn 12225 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
83 2cnd 12295 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 2 ∈ β„‚)
8482, 83pncand 11577 . . . . . . . . . . . . 13 (𝑁 ∈ β„• β†’ ((𝑁 + 2) βˆ’ 2) = 𝑁)
85843ad2ant3 1134 . . . . . . . . . . . 12 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((𝑁 + 2) βˆ’ 2) = 𝑁)
8685ad2antrr 723 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ ((𝑁 + 2) βˆ’ 2) = 𝑁)
8786fveq2d 6895 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) = ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘))
8887neeq1d 2999 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0)))
8988anbi2d 628 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ ((((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0)) ↔ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜π‘) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0))))
9081, 89mpbird 257 . . . . . . 7 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0)))
9190biantrud 531 . . . . . 6 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0)))))
9261anim2i 616 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (β„€β‰₯β€˜2)))
93923adant1 1129 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (β„€β‰₯β€˜2)))
9493ad2antrr 723 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (β„€β‰₯β€˜2)))
95 numclwwlk.h . . . . . . . . . 10 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})
9695numclwwlkovh 29894 . . . . . . . . 9 ((𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐻(𝑁 + 2)) = {𝑀 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜((𝑁 + 2) βˆ’ 2)) β‰  (π‘€β€˜0))})
9794, 96syl 17 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (𝑋𝐻(𝑁 + 2)) = {𝑀 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜((𝑁 + 2) βˆ’ 2)) β‰  (π‘€β€˜0))})
9897eleq2d 2818 . . . . . . 7 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2)) ↔ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ {𝑀 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜((𝑁 + 2) βˆ’ 2)) β‰  (π‘€β€˜0))}))
99 fveq1 6890 . . . . . . . . . 10 (𝑀 = (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) β†’ (π‘€β€˜0) = ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0))
10099eqeq1d 2733 . . . . . . . . 9 (𝑀 = (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) β†’ ((π‘€β€˜0) = 𝑋 ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋))
101 fveq1 6890 . . . . . . . . . 10 (𝑀 = (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) β†’ (π‘€β€˜((𝑁 + 2) βˆ’ 2)) = ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)))
102101, 99neeq12d 3001 . . . . . . . . 9 (𝑀 = (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) β†’ ((π‘€β€˜((𝑁 + 2) βˆ’ 2)) β‰  (π‘€β€˜0) ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0)))
103100, 102anbi12d 630 . . . . . . . 8 (𝑀 = (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) β†’ (((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜((𝑁 + 2) βˆ’ 2)) β‰  (π‘€β€˜0)) ↔ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0))))
104103elrab 3683 . . . . . . 7 ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ {𝑀 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜((𝑁 + 2) βˆ’ 2)) β‰  (π‘€β€˜0))} ↔ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0))))
10598, 104bitr2di 288 . . . . . 6 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0) = 𝑋 ∧ ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜((𝑁 + 2) βˆ’ 2)) β‰  ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0))) ↔ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2))))
10667, 91, 1053bitrd 305 . . . . 5 ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) ∧ 𝑣 ∈ 𝑉) β†’ (({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ↔ (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2))))
107106reubidva 3391 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ (βˆƒ!𝑣 ∈ 𝑉 ({(lastSβ€˜π‘Š), 𝑣} ∈ (Edgβ€˜πΊ) ∧ {𝑣, (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ↔ βˆƒ!𝑣 ∈ 𝑉 (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2))))
10842, 107mpbid 231 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋))) β†’ βˆƒ!𝑣 ∈ 𝑉 (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2)))
109108ex 412 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ βˆƒ!𝑣 ∈ 𝑉 (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2))))
11012, 109sylbid 239 1 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ (𝑋𝑄𝑁) β†’ βˆƒ!𝑣 ∈ 𝑉 (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒ!wreu 3373  {crab 3431  Vcvv 3473  {cpr 4630  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  0cc0 11114  1c1 11115   + caddc 11117   βˆ’ cmin 11449  β„•cn 12217  2c2 12272  β„•0cn0 12477  β„€cz 12563  β„€β‰₯cuz 12827  ..^cfzo 13632  β™―chash 14295  Word cword 14469  lastSclsw 14517   ++ cconcat 14525  βŸ¨β€œcs1 14550  Vtxcvtx 28524  Edgcedg 28575   WWalksN cwwlksn 29348   ClWWalksN cclwwlkn 29545  ClWWalksNOncclwwlknon 29608   FriendGraph cfrgr 29779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-oadd 8474  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-rp 12980  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-lsw 14518  df-concat 14526  df-s1 14551  df-wwlks 29352  df-wwlksn 29353  df-clwwlk 29503  df-clwwlkn 29546  df-clwwlknon 29609  df-frgr 29780
This theorem is referenced by:  numclwlk2lem2f1o  29900
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