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 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
numclwwlk.g	⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
numclwwlk.h	⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})

Assertion

Ref	Expression
numclwwlk2lem1	⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (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.

Step	Hyp	Ref	Expression
1		nnnn0 11176	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2	1	anim2i 591	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0))
3	2	3adant1 1072	. . . . 5 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0))
4		numclwwlk.c	. . . . . 6 ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
5		numclwwlk.f	. . . . . 6 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
6		numclwwlk.g	. . . . . 6 ⊢ 𝐺 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
7		numclwwlk.q	. . . . . 6 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
8	4, 5, 6, 7	numclwwlkovq 26626	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
9	3, 8	syl 17	. . . 4 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
10	9	eleq2d 2673	. . 3 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 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))
12	11	eqeq1d 2612	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
13		fveq2 6103	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( lastS ‘𝑤) = ( lastS ‘𝑊))
14	13	neeq1d 2841	. . . . 5 ⊢ (𝑤 = 𝑊 → (( lastS ‘𝑤) ≠ 𝑋 ↔ ( lastS ‘𝑊) ≠ 𝑋))
15	12, 14	anbi12d 743	. . . 4 ⊢ (𝑤 = 𝑊 → (((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋) ↔ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)))
16	15	elrab 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)))
17	10, 16	syl6bb 275	. 2 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))))
18		simpl1 1057	. . . . 5 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 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){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
20		peano2nn 10909	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
21	20	adantl 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)))
23	21, 22	jca 553	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))))
24	23	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))))
25	24	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))))
26	19, 25	syl 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)
28	26, 27	syl6 34	. . . . . . . . . 10 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑁 ∈ ℕ → ( lastS ‘𝑊) ∈ 𝑉))
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑁 ∈ ℕ → ( lastS ‘𝑊) ∈ 𝑉))
30	29	com12 32	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ∈ 𝑉))
31	30	3ad2ant3 1077	. . . . . . 7 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ∈ 𝑉))
32	31	imp 444	. . . . . 6 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ( lastS ‘𝑊) ∈ 𝑉)
33		eleq1 2676	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
34	33	biimprd 237	. . . . . . . . . 10 ⊢ ((𝑊‘0) = 𝑋 → (𝑋 ∈ 𝑉 → (𝑊‘0) ∈ 𝑉))
35	34	ad2antrl 760	. . . . . . . . 9 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑋 ∈ 𝑉 → (𝑊‘0) ∈ 𝑉))
36	35	com12 32	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
37	36	3ad2ant2 1076	. . . . . . 7 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
38	37	imp 444	. . . . . 6 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 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)))
40	39	eqcoms 2618	. . . . . . . . 9 ⊢ ((𝑊‘0) = 𝑋 → (( lastS ‘𝑊) ≠ 𝑋 ↔ ( lastS ‘𝑊) ≠ (𝑊‘0)))
41	40	biimpa 500	. . . . . . . 8 ⊢ (((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋) → ( lastS ‘𝑊) ≠ (𝑊‘0))
42	41	adantl 481	. . . . . . 7 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ≠ (𝑊‘0))
43	42	adantl 481	. . . . . 6 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ( lastS ‘𝑊) ≠ (𝑊‘0))
44	32, 38, 43	3jca 1235	. . . . 5 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 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)) → ∃!𝑣 ∈ 𝑉 ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)))
46	18, 44, 45	sylc 63	. . . 4 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ∃!𝑣 ∈ 𝑉 ({( 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))
48	47	ad2antlr 759	. . . . . . . . . 10 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
49		simpr 476	. . . . . . . . . 10 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
50	1	3ad2ant3 1077	. . . . . . . . . . 11 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
51	50	ad2antrr 758	. . . . . . . . . 10 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑁 ∈ ℕ0)
52	48, 49, 51	3jca 1235	. . . . . . . . 9 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0))
53		wwlkext2clwwlk 26331	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
54	53	imp 444	. . . . . . . . 9 ⊢ (((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
55	52, 54	sylan 487	. . . . . . . 8 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
56		2nn0 11186	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ0
57	56	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ0)
58	1, 57	nn0addcld 11232	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ ℕ0)
59	4	numclwwlkfvc 26604	. . . . . . . . . . 11 ⊢ ((𝑁 + 2) ∈ ℕ0 → (𝐶‘(𝑁 + 2)) = ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
60	58, 59	syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝐶‘(𝑁 + 2)) = ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
61	60	3ad2ant3 1077	. . . . . . . . 9 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝐶‘(𝑁 + 2)) = ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
62	61	ad3antrrr 762	. . . . . . . 8 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸)) → (𝐶‘(𝑁 + 2)) = ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
63	55, 62	eleqtrrd 2691	. . . . . . 7 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ ({( 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 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)))
65	64	simprrd 793	. . . . . . . . . 10 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ Word 𝑉)
66	65	ad2antrl 760	. . . . . . . . 9 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
67	66	ad2antrr 758	. . . . . . . 8 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → 𝑊 ∈ Word 𝑉)
68	49	adantr 480	. . . . . . . 8 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → 𝑣 ∈ 𝑉)
69		2z 11286	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
70		nn0pzuz 11621	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ≥‘2))
71	1, 69, 70	sylancl 693	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ≥‘2))
72	71	3ad2ant3 1077	. . . . . . . . 9 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 2) ∈ (ℤ≥‘2))
73	72	ad3antrrr 762	. . . . . . . 8 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → (𝑁 + 2) ∈ (ℤ≥‘2))
74	59	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝑁 + 2) ∈ ℕ0 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
75	58, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
76	75	3ad2ant3 1077	. . . . . . . . . 10 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
77	76	ad2antrr 758	. . . . . . . . 9 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))
78	77	biimpa 500	. . . . . . . 8 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2)))
79		clwwlkext2edg 26330	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑣 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ≥‘2)) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))) → ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸))
80	67, 68, 73, 78, 79	syl31anc 1321	. . . . . . 7 ⊢ (((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))) → ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸))
81	63, 80	impbida 873	. . . . . 6 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2))))
82		df-3an 1033	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑣 ∈ 𝑉) ↔ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑣 ∈ 𝑉))
83	82	simplbi2 653	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑣 ∈ 𝑉 → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑣 ∈ 𝑉)))
84	83	3adant1 1072	. . . . . . . . . . . 12 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑣 ∈ 𝑉 → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑣 ∈ 𝑉)))
85	84	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (𝑣 ∈ 𝑉 → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑣 ∈ 𝑉)))
86	85	imp 444	. . . . . . . . . 10 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑣 ∈ 𝑉))
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)))
88	87	biimpri 217	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))
89	88	ad2antlr 759	. . . . . . . . . 10 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))
90	86, 89	jca 553	. . . . . . . . 9 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)))
91		clwwlkextfrlem1 26603	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑣 ∈ 𝑉) ∧ (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)))
94	93	eqcoms 2618	. . . . . . . . . . 11 ⊢ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
95	94	biimpa 500	. . . . . . . . . 10 ⊢ ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋) → ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))
96	92, 95	jca 553	. . . . . . . . 9 ⊢ ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ 𝑋) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
97	90, 91, 96	3syl 18	. . . . . . . 8 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((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)
100	98, 99	pncand 10272	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 2) − 2) = 𝑁)
101	100	3ad2ant3 1077	. . . . . . . . . . . 12 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑁 + 2) − 2) = 𝑁)
102	101	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑁 + 2) − 2) = 𝑁)
103	102	fveq2d 6107	. . . . . . . . . 10 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁))
104	103	neeq1d 2841	. . . . . . . . 9 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
105	104	anbi2d 736	. . . . . . . 8 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)) ↔ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))))
106	97, 105	mpbird 246	. . . . . . 7 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
107	106	biantrud 527	. . . . . 6 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((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)
109	1, 56, 108	sylancl 693	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ ℕ0)
110	109	anim2i 591	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ0))
111	110	3adant1 1072	. . . . . . . . . 10 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ0))
112	111	ad2antrr 758	. . . . . . . . 9 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ0))
113		numclwwlk.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
114	4, 5, 6, 7, 113	numclwwlkovh 26628	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ0) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ (𝐶‘(𝑁 + 2)) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
115	112, 114	syl 17	. . . . . . . 8 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ (𝐶‘(𝑁 + 2)) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
116	115	eleq2d 2673	. . . . . . 7 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((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))
118	117	eqeq1d 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)))
120	119, 117	neeq12d 2843	. . . . . . . . 9 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
121	118, 120	anbi12d 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))))
122	121	elrab 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))))
123	116, 122	syl6rbb 276	. . . . . 6 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝐶‘(𝑁 + 2)) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
124	81, 107, 123	3bitrd 293	. . . . 5 ⊢ ((((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
125	124	reubidva 3102	. . . 4 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (∃!𝑣 ∈ 𝑉 ({( lastS ‘𝑊), 𝑣} ∈ ran 𝐸 ∧ {𝑣, (𝑊‘0)} ∈ ran 𝐸) ↔ ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
126	46, 125	mpbid 221	. . 3 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
127	126	ex 449	. 2 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
128	17, 127	sylbid 229	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))

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