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Theorem numclwwlk2lem1 29629
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 29627 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
433adant1 1131 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
54eleq2d 2820 . . 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 6891 . . . . . 6 (𝑀 = π‘Š β†’ (π‘€β€˜0) = (π‘Šβ€˜0))
76eqeq1d 2735 . . . . 5 (𝑀 = π‘Š β†’ ((π‘€β€˜0) = 𝑋 ↔ (π‘Šβ€˜0) = 𝑋))
8 fveq2 6892 . . . . . 6 (𝑀 = π‘Š β†’ (lastSβ€˜π‘€) = (lastSβ€˜π‘Š))
98neeq1d 3001 . . . . 5 (𝑀 = π‘Š β†’ ((lastSβ€˜π‘€) β‰  𝑋 ↔ (lastSβ€˜π‘Š) β‰  𝑋))
107, 9anbi12d 632 . . . 4 (𝑀 = π‘Š β†’ (((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋) ↔ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)))
1110elrab 3684 . . 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 1192 . . . . 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 2733 . . . . . . . . . . . . 13 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
151, 14wwlknp 29097 . . . . . . . . . . . 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 12224 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„• β†’ (𝑁 + 1) ∈ β„•)
1716adantl 483 . . . . . . . . . . . . . . 15 (((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑁 ∈ β„•) β†’ (𝑁 + 1) ∈ β„•)
18 simpl 484 . . . . . . . . . . . . . . 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 513 . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . 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 1133 . . . . . . . . . . . 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 14519 . . . . . . . . . . 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 482 . . . . . . . . 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 1136 . . . . . . 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 408 . . . . . 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 2822 . . . . . . . . . . 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 727 . . . . . . . . 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 1135 . . . . . . 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 408 . . . . . 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 3005 . . . . . . . . . 10 (𝑋 = (π‘Šβ€˜0) β†’ ((lastSβ€˜π‘Š) β‰  𝑋 ↔ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)))
3635eqcoms 2741 . . . . . . . . 9 ((π‘Šβ€˜0) = 𝑋 β†’ ((lastSβ€˜π‘Š) β‰  𝑋 ↔ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)))
3736biimpa 478 . . . . . . . 8 (((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋) β†’ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))
3837adantl 483 . . . . . . 7 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))
3938adantl 483 . . . . . 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 1129 . . . . 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 29520 . . . . 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 484 . . . . . . . . . 10 ((π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ ((π‘Šβ€˜0) = 𝑋 ∧ (lastSβ€˜π‘Š) β‰  𝑋)) β†’ π‘Š ∈ (𝑁 WWalksN 𝐺))
4443ad2antlr 726 . . . . . . . . 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 486 . . . . . . . . 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 12479 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„•0)
47463ad2ant3 1136 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ β„•0)
4847ad2antrr 725 . . . . . . . . 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 1129 . . . . . . . 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 29310 . . . . . . . . . 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 1133 . . . . . . . . 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 408 . . . . . . . 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 581 . . . . . . 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 29096 . . . . . . . . . . 11 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉))
5554simp3d 1145 . . . . . . . . . 10 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ π‘Š ∈ Word 𝑉)
5655ad2antrl 727 . . . . . . . . 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 725 . . . . . . . 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 482 . . . . . . . 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 12594 . . . . . . . . . . 11 2 ∈ β„€
60 nn0pzuz 12889 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ 2 ∈ β„€) β†’ (𝑁 + 2) ∈ (β„€β‰₯β€˜2))
6146, 59, 60sylancl 587 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (𝑁 + 2) ∈ (β„€β‰₯β€˜2))
62613ad2ant3 1136 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑁 + 2) ∈ (β„€β‰₯β€˜2))
6362ad3antrrr 729 . . . . . . . 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 486 . . . . . . . 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 29309 . . . . . . . 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 1374 . . . . . . 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 800 . . . . . 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 2844 . . . . . . . . . 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 618 . . . . . . . . . . . 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 726 . . . . . . . . . . 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 497 . . . . . . . . . 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 29595 . . . . . . . . . 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 1372 . . . . . . . . 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 2745 . . . . . . . . . . . . 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 2741 . . . . . . . . . . . 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 727 . . . . . . . . . . 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 726 . . . . . . . . . 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 496 . . . . . . . . . . 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 3002 . . . . . . . . . 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 632 . . . . . . . . 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 12220 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
83 2cnd 12290 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 2 ∈ β„‚)
8482, 83pncand 11572 . . . . . . . . . . . . 13 (𝑁 ∈ β„• β†’ ((𝑁 + 2) βˆ’ 2) = 𝑁)
85843ad2ant3 1136 . . . . . . . . . . . 12 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((𝑁 + 2) βˆ’ 2) = 𝑁)
8685ad2antrr 725 . . . . . . . . . . 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 6896 . . . . . . . . . 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 3001 . . . . . . . . 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 630 . . . . . . . 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 533 . . . . . 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 618 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (β„€β‰₯β€˜2)))
93923adant1 1131 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (β„€β‰₯β€˜2)))
9493ad2antrr 725 . . . . . . . . 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 29626 . . . . . . . . 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 2820 . . . . . . 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 6891 . . . . . . . . . 10 (𝑀 = (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) β†’ (π‘€β€˜0) = ((π‘Š ++ βŸ¨β€œπ‘£β€βŸ©)β€˜0))
10099eqeq1d 2735 . . . . . . . . 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 6891 . . . . . . . . . 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 3003 . . . . . . . . 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 632 . . . . . . . 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 3684 . . . . . . 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 3393 . . . 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 414 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒ!wreu 3375  {crab 3433  Vcvv 3475  {cpr 4631  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  0cc0 11110  1c1 11111   + caddc 11113   βˆ’ cmin 11444  β„•cn 12212  2c2 12267  β„•0cn0 12472  β„€cz 12558  β„€β‰₯cuz 12822  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512   ++ cconcat 14520  βŸ¨β€œcs1 14545  Vtxcvtx 28256  Edgcedg 28307   WWalksN cwwlksn 29080   ClWWalksN cclwwlkn 29277  ClWWalksNOncclwwlknon 29340   FriendGraph cfrgr 29511
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-concat 14521  df-s1 14546  df-wwlks 29084  df-wwlksn 29085  df-clwwlk 29235  df-clwwlkn 29278  df-clwwlknon 29341  df-frgr 29512
This theorem is referenced by:  numclwlk2lem2f1o  29632
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