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Theorem numclwwlk2lem1 26629
 Description: In a friendship graph, for each walk of length n starting with a fixed vertex 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 only generally holds for Friendship Graphs, because these guarantee that for the first and last vertex there is a third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.)
Hypotheses
Ref Expression
numclwwlk.c 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
numclwwlk.h 𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
Assertion
Ref Expression
numclwwlk2lem1 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉   𝑤,𝐸   𝑤,𝑉   𝑤,𝐹   𝑤,𝑄   𝑤,𝐺   𝑣,𝐸   𝑣,𝑊,𝑤
Allowed substitution hints:   𝑄(𝑣,𝑛)   𝐹(𝑣,𝑛)   𝐺(𝑣,𝑛)   𝐻(𝑤,𝑣,𝑛)   𝑊(𝑛)

Proof of Theorem numclwwlk2lem1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 nnnn0 11176 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
21anim2i 591 . . . . . 6 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉𝑁 ∈ ℕ0))
323adant1 1072 . . . . 5 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉𝑁 ∈ ℕ0))
4 numclwwlk.c . . . . . 6 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
5 numclwwlk.f . . . . . 6 𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})
6 numclwwlk.g . . . . . 6 𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
7 numclwwlk.q . . . . . 6 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
84, 5, 6, 7numclwwlkovq 26626 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
93, 8syl 17 . . . 4 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
109eleq2d 2673 . . 3 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ 𝑊 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}))
11 fveq1 6102 . . . . . 6 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
1211eqeq1d 2612 . . . . 5 (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
13 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ( lastS ‘𝑤) = ( lastS ‘𝑊))
1413neeq1d 2841 . . . . 5 (𝑤 = 𝑊 → (( lastS ‘𝑤) ≠ 𝑋 ↔ ( lastS ‘𝑊) ≠ 𝑋))
1512, 14anbi12d 743 . . . 4 (𝑤 = 𝑊 → (((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋) ↔ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)))
1615elrab 3331 . . 3 (𝑊 ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} ↔ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)))
1710, 16syl6bb 275 . 2 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))))
18 simpl1 1057 . . . . 5 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 𝑉 FriendGrph 𝐸)
19 wwlknimp 26215 . . . . . . . . . . . 12 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
20 peano2nn 10909 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
2120adantl 481 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
22 simpl 472 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))
2321, 22jca 553 . . . . . . . . . . . . . 14 (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))))
2423ex 449 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))))
25243adant3 1074 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))))
2619, 25syl 17 . . . . . . . . . . 11 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))))
27 lswlgt0cl 13209 . . . . . . . . . . 11 (((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → ( lastS ‘𝑊) ∈ 𝑉)
2826, 27syl6 34 . . . . . . . . . 10 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ → ( lastS ‘𝑊) ∈ 𝑉))
2928adantr 480 . . . . . . . . 9 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑁 ∈ ℕ → ( lastS ‘𝑊) ∈ 𝑉))
3029com12 32 . . . . . . . 8 (𝑁 ∈ ℕ → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ∈ 𝑉))
31303ad2ant3 1077 . . . . . . 7 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ∈ 𝑉))
3231imp 444 . . . . . 6 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ( lastS ‘𝑊) ∈ 𝑉)
33 eleq1 2676 . . . . . . . . . . 11 ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ 𝑉𝑋𝑉))
3433biimprd 237 . . . . . . . . . 10 ((𝑊‘0) = 𝑋 → (𝑋𝑉 → (𝑊‘0) ∈ 𝑉))
3534ad2antrl 760 . . . . . . . . 9 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑋𝑉 → (𝑊‘0) ∈ 𝑉))
3635com12 32 . . . . . . . 8 (𝑋𝑉 → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
37363ad2ant2 1076 . . . . . . 7 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
3837imp 444 . . . . . 6 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (𝑊‘0) ∈ 𝑉)
39 neeq2 2845 . . . . . . . . . 10 (𝑋 = (𝑊‘0) → (( lastS ‘𝑊) ≠ 𝑋 ↔ ( lastS ‘𝑊) ≠ (𝑊‘0)))
4039eqcoms 2618 . . . . . . . . 9 ((𝑊‘0) = 𝑋 → (( lastS ‘𝑊) ≠ 𝑋 ↔ ( lastS ‘𝑊) ≠ (𝑊‘0)))
4140biimpa 500 . . . . . . . 8 (((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋) → ( lastS ‘𝑊) ≠ (𝑊‘0))
4241adantl 481 . . . . . . 7 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ≠ (𝑊‘0))
4342adantl 481 . . . . . 6 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ( lastS ‘𝑊) ≠ (𝑊‘0))
4432, 38, 433jca 1235 . . . . 5 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (( lastS ‘𝑊) ∈ 𝑉 ∧ (𝑊‘0) ∈ 𝑉 ∧ ( lastS ‘𝑊) ≠ (𝑊‘0)))
45 frgraun 26523 . . . . 5 (𝑉 FriendGrph 𝐸 → ((( lastS ‘𝑊) ∈ 𝑉 ∧ (𝑊‘0) ∈ 𝑉 ∧ ( lastS ‘𝑊) ≠ (𝑊‘0)) → ∃!𝑣𝑉 ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)))
4618, 44, 45sylc 63 . . . 4 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ∃!𝑣𝑉 ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸))
47 simpl 472 . . . . . . . . . . 11 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
4847ad2antlr 759 . . . . . . . . . 10 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
49 simpr 476 . . . . . . . . . 10 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑣𝑉)
5013ad2ant3 1077 . . . . . . . . . . 11 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
5150ad2antrr 758 . . . . . . . . . 10 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑁 ∈ ℕ0)
5248, 49, 513jca 1235 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑣𝑉𝑁 ∈ ℕ0))
53 wwlkext2clwwlk 26331 . . . . . . . . . 10 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑣𝑉𝑁 ∈ ℕ0) → (({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
5453imp 444 . . . . . . . . 9 (((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑣𝑉𝑁 ∈ ℕ0) ∧ ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
5552, 54sylan 487 . . . . . . . 8 (((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
56 2nn0 11186 . . . . . . . . . . . . 13 2 ∈ ℕ0
5756a1i 11 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 2 ∈ ℕ0)
581, 57nn0addcld 11232 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑁 + 2) ∈ ℕ0)
594numclwwlkfvc 26604 . . . . . . . . . . 11 ((𝑁 + 2) ∈ ℕ0 → (𝐶‘(𝑁 + 2)) = ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
6058, 59syl 17 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝐶‘(𝑁 + 2)) = ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
61603ad2ant3 1077 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝐶‘(𝑁 + 2)) = ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
6261ad3antrrr 762 . . . . . . . 8 (((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)) → (𝐶‘(𝑁 + 2)) = ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
6355, 62eleqtrrd 2691 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)))
64 wwlknprop 26214 . . . . . . . . . . 11 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉)))
6564simprrd 793 . . . . . . . . . 10 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ Word 𝑉)
6665ad2antrl 760 . . . . . . . . 9 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
6766ad2antrr 758 . . . . . . . 8 (((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → 𝑊 ∈ Word 𝑉)
6849adantr 480 . . . . . . . 8 (((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → 𝑣𝑉)
69 2z 11286 . . . . . . . . . . 11 2 ∈ ℤ
70 nn0pzuz 11621 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ‘2))
711, 69, 70sylancl 693 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ‘2))
72713ad2ant3 1077 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑁 + 2) ∈ (ℤ‘2))
7372ad3antrrr 762 . . . . . . . 8 (((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → (𝑁 + 2) ∈ (ℤ‘2))
7459eleq2d 2673 . . . . . . . . . . . 12 ((𝑁 + 2) ∈ ℕ0 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
7558, 74syl 17 . . . . . . . . . . 11 (𝑁 ∈ ℕ → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
76753ad2ant3 1077 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
7776ad2antrr 758 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
7877biimpa 500 . . . . . . . 8 (((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
79 clwwlkext2edg 26330 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑣𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))) → ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸))
8067, 68, 73, 78, 79syl31anc 1321 . . . . . . 7 (((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸))
8163, 80impbida 873 . . . . . 6 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))))
82 df-3an 1033 . . . . . . . . . . . . . 14 ((𝑋𝑉𝑁 ∈ ℕ ∧ 𝑣𝑉) ↔ ((𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑣𝑉))
8382simplbi2 653 . . . . . . . . . . . . 13 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑣𝑉 → (𝑋𝑉𝑁 ∈ ℕ ∧ 𝑣𝑉)))
84833adant1 1072 . . . . . . . . . . . 12 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑣𝑉 → (𝑋𝑉𝑁 ∈ ℕ ∧ 𝑣𝑉)))
8584adantr 480 . . . . . . . . . . 11 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (𝑣𝑉 → (𝑋𝑉𝑁 ∈ ℕ ∧ 𝑣𝑉)))
8685imp 444 . . . . . . . . . 10 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑋𝑉𝑁 ∈ ℕ ∧ 𝑣𝑉))
87 3anass 1035 . . . . . . . . . . . 12 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋) ↔ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)))
8887biimpri 217 . . . . . . . . . . 11 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))
8988ad2antlr 759 . . . . . . . . . 10 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))
9086, 89jca 553 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑋𝑉𝑁 ∈ ℕ ∧ 𝑣𝑉) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)))
91 clwwlkextfrlem1 26603 . . . . . . . . 9 (((𝑋𝑉𝑁 ∈ ℕ ∧ 𝑣𝑉) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋))
92 simpl 472 . . . . . . . . . 10 ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋) → ((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋)
93 neeq2 2845 . . . . . . . . . . . 12 (𝑋 = ((𝑊 ++ ⟨“𝑣”⟩)‘0) → (((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
9493eqcoms 2618 . . . . . . . . . . 11 (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
9594biimpa 500 . . . . . . . . . 10 ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋) → ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))
9692, 95jca 553 . . . . . . . . 9 ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
9790, 91, 963syl 18 . . . . . . . 8 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
98 nncn 10905 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
99 2cnd 10970 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 2 ∈ ℂ)
10098, 99pncand 10272 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → ((𝑁 + 2) − 2) = 𝑁)
1011003ad2ant3 1077 . . . . . . . . . . . 12 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → ((𝑁 + 2) − 2) = 𝑁)
102101ad2antrr 758 . . . . . . . . . . 11 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑁 + 2) − 2) = 𝑁)
103102fveq2d 6107 . . . . . . . . . 10 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁))
104103neeq1d 2841 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
105104anbi2d 736 . . . . . . . 8 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)) ↔ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))))
10697, 105mpbird 246 . . . . . . 7 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
107106biantrud 527 . . . . . 6 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))))
108 nn0addcl 11205 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ 2 ∈ ℕ0) → (𝑁 + 2) ∈ ℕ0)
1091, 56, 108sylancl 693 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → (𝑁 + 2) ∈ ℕ0)
110109anim2i 591 . . . . . . . . . . 11 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ ℕ0))
1111103adant1 1072 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ ℕ0))
112111ad2antrr 758 . . . . . . . . 9 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ ℕ0))
113 numclwwlk.h . . . . . . . . . 10 𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
1144, 5, 6, 7, 113numclwwlkovh 26628 . . . . . . . . 9 ((𝑋𝑉 ∧ (𝑁 + 2) ∈ ℕ0) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ (𝐶‘(𝑁 + 2)) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
115112, 114syl 17 . . . . . . . 8 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ (𝐶‘(𝑁 + 2)) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
116115eleq2d 2673 . . . . . . 7 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ {𝑤 ∈ (𝐶‘(𝑁 + 2)) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))}))
117 fveq1 6102 . . . . . . . . . 10 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘0) = ((𝑊 ++ ⟨“𝑣”⟩)‘0))
118117eqeq1d 2612 . . . . . . . . 9 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋))
119 fveq1 6102 . . . . . . . . . 10 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)))
120119, 117neeq12d 2843 . . . . . . . . 9 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
121118, 120anbi12d 743 . . . . . . . 8 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0)) ↔ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))))
122121elrab 3331 . . . . . . 7 ((𝑊 ++ ⟨“𝑣”⟩) ∈ {𝑤 ∈ (𝐶‘(𝑁 + 2)) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))} ↔ ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))))
123116, 122syl6rbb 276 . . . . . 6 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
12481, 107, 1233bitrd 293 . . . . 5 ((((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
125124reubidva 3102 . . . 4 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (∃!𝑣𝑉 ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸) ↔ ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
12646, 125mpbid 221 . . 3 (((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
127126ex 449 . 2 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
12817, 127sylbid 229 1 ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃!wreu 2898  {crab 2900  Vcvv 3173  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   WWalksN cwwlkn 26206   ClWWalksN cclwwlkn 26277   FriendGrph cfrgra 26515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280  df-frgra 26516 This theorem is referenced by:  numclwlk2lem2f1o  26632
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