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Theorem numclwwlk2lem1 27235

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.)

**Hypotheses**
Ref	Expression
numclwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
numclwwlk.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
numclwwlk.h	⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})

**Assertion**
Ref	Expression
numclwwlk2lem1	⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))

Distinct variable groups: 𝑛,𝐺,𝑣,𝑤 𝑛,𝑁,𝑣,𝑤 𝑛,𝑉,𝑣 𝑛,𝑋,𝑣,𝑤 𝑤,𝑉 𝑣,𝑊,𝑤 Allowed substitution hints: 𝑄(𝑤,𝑣,𝑛) 𝐹(𝑤,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		numclwwlk.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2		numclwwlk.q	. . . . . 6 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
3	1, 2	numclwwlkovq 27232	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
4	3	3adant1 1079	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})
5	4	eleq2d 2687	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}))
6		fveq1 6190	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
7	6	eqeq1d 2624	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
8		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( lastS ‘𝑤) = ( lastS ‘𝑊))
9	8	neeq1d 2853	. . . . 5 ⊢ (𝑤 = 𝑊 → (( lastS ‘𝑤) ≠ 𝑋 ↔ ( lastS ‘𝑊) ≠ 𝑋))
10	7, 9	anbi12d 747	. . . 4 ⊢ (𝑤 = 𝑊 → (((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋) ↔ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)))
11	10	elrab 3363	. . 3 ⊢ (𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)))
12	5, 11	syl6bb 276	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))))
13		simpl1 1064	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 𝐺 ∈ FriendGraph )
14		eqid 2622	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
15	1, 14	wwlknp 26734	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
16		peano2nn 11032	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
17	16	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
18		simpl 473	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))
19	17, 18	jca 554	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))))
20	19	ex 450	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))))
21	20	3adant3 1081	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))))
22	15, 21	syl 17	. . . . . . . . . . 11 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))))
23		lswlgt0cl 13356	. . . . . . . . . . 11 ⊢ (((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1))) → ( lastS ‘𝑊) ∈ 𝑉)
24	22, 23	syl6 35	. . . . . . . . . 10 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → ( lastS ‘𝑊) ∈ 𝑉))
25	24	adantr 481	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑁 ∈ ℕ → ( lastS ‘𝑊) ∈ 𝑉))
26	25	com12 32	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ∈ 𝑉))
27	26	3ad2ant3 1084	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ∈ 𝑉))
28	27	imp 445	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ( lastS ‘𝑊) ∈ 𝑉)
29		eleq1 2689	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
30	29	biimprd 238	. . . . . . . . . 10 ⊢ ((𝑊‘0) = 𝑋 → (𝑋 ∈ 𝑉 → (𝑊‘0) ∈ 𝑉))
31	30	ad2antrl 764	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑋 ∈ 𝑉 → (𝑊‘0) ∈ 𝑉))
32	31	com12 32	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
33	32	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
34	33	imp 445	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (𝑊‘0) ∈ 𝑉)
35		neeq2 2857	. . . . . . . . . 10 ⊢ (𝑋 = (𝑊‘0) → (( lastS ‘𝑊) ≠ 𝑋 ↔ ( lastS ‘𝑊) ≠ (𝑊‘0)))
36	35	eqcoms 2630	. . . . . . . . 9 ⊢ ((𝑊‘0) = 𝑋 → (( lastS ‘𝑊) ≠ 𝑋 ↔ ( lastS ‘𝑊) ≠ (𝑊‘0)))
37	36	biimpa 501	. . . . . . . 8 ⊢ (((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋) → ( lastS ‘𝑊) ≠ (𝑊‘0))
38	37	adantl 482	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ( lastS ‘𝑊) ≠ (𝑊‘0))
39	38	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ( lastS ‘𝑊) ≠ (𝑊‘0))
40	28, 34, 39	3jca 1242	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (( lastS ‘𝑊) ∈ 𝑉 ∧ (𝑊‘0) ∈ 𝑉 ∧ ( lastS ‘𝑊) ≠ (𝑊‘0)))
41	1, 14	frcond2 27131	. . . . 5 ⊢ (𝐺 ∈ FriendGraph → ((( lastS ‘𝑊) ∈ 𝑉 ∧ (𝑊‘0) ∈ 𝑉 ∧ ( lastS ‘𝑊) ≠ (𝑊‘0)) → ∃!𝑣 ∈ 𝑉 ({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺))))
42	13, 40, 41	sylc 65	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ∃!𝑣 ∈ 𝑉 ({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)))
43		simpl 473	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
44	43	ad2antlr 763	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
45		simpr 477	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
46		nnnn0 11299	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
47	46	3ad2ant3 1084	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
48	47	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑁 ∈ ℕ₀)
49	44, 45, 48	3jca 1242	. . . . . . . 8 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀))
50	1, 14	wwlksext2clwwlk 26924	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
51	50	imp 445	. . . . . . . 8 ⊢ (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) ∧ ({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
52	49, 51	sylan 488	. . . . . . 7 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ ({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
53	1	wwlknbp 26733	. . . . . . . . . . 11 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉))
54	53	simp3d 1075	. . . . . . . . . 10 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ Word 𝑉)
55	54	ad2antrl 764	. . . . . . . . 9 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
56	55	ad2antrr 762	. . . . . . . 8 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → 𝑊 ∈ Word 𝑉)
57	45	adantr 481	. . . . . . . 8 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → 𝑣 ∈ 𝑉)
58		2z 11409	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
59		nn0pzuz 11745	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ_≥‘2))
60	46, 58, 59	sylancl 694	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ_≥‘2))
61	60	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 2) ∈ (ℤ_≥‘2))
62	61	ad3antrrr 766	. . . . . . . 8 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → (𝑁 + 2) ∈ (ℤ_≥‘2))
63		simpr 477	. . . . . . . 8 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
64	1, 14	clwwlksext2edg 26923	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑣 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → ({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)))
65	56, 57, 62, 63, 64	syl31anc 1329	. . . . . . 7 ⊢ (((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → ({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)))
66	52, 65	impbida 877	. . . . . 6 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
67	47	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → 𝑁 ∈ ℕ₀)
68	1	eleq2i 2693	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtx‘𝐺))
69	68	biimpi 206	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝑉 → 𝑣 ∈ (Vtx‘𝐺))
70	67, 69	anim12i 590	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑁 ∈ ℕ₀ ∧ 𝑣 ∈ (Vtx‘𝐺)))
71	37	anim2i 593	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑊) ≠ (𝑊‘0)))
72	71	ad2antlr 763	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑊) ≠ (𝑊‘0)))
73		clwwlkextfrlem1 27208	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑣 ∈ (Vtx‘𝐺)) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑊) ≠ (𝑊‘0))) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0)))
74	70, 72, 73	syl2anc 693	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0)))
75		eqeq2 2633	. . . . . . . . . . . . 13 ⊢ (𝑋 = (𝑊‘0) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
76	75	eqcoms 2630	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
77	76	ad2antrl 764	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
78	77	ad2antlr 763	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
79	74	simpld 475	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0))
80	79	neeq2d 2854	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0)))
81	78, 80	anbi12d 747	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)) ↔ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0))))
82	74, 81	mpbird 247	. . . . . . . 8 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
83		nncn 11028	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
84		2cnd 11093	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
85	83, 84	pncand 10393	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 2) − 2) = 𝑁)
86	85	3ad2ant3 1084	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑁 + 2) − 2) = 𝑁)
87	86	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑁 + 2) − 2) = 𝑁)
88	87	fveq2d 6195	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁))
89	88	neeq1d 2853	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
90	89	anbi2d 740	. . . . . . . 8 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((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))))
91	82, 90	mpbird 247	. . . . . . 7 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
92	91	biantrud 528	. . . . . 6 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ↔ ((𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))))
93		2nn 11185	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
94		nnaddcl 11042	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 2 ∈ ℕ) → (𝑁 + 2) ∈ ℕ)
95	93, 94	mpan2 707	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ ℕ)
96	95	anim2i 593	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ))
97	96	3adant1 1079	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ))
98	97	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ))
99		numclwwlk.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
100		numclwwlk.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})
101	1, 2, 99, 100	numclwwlkovh 27234	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ ℕ) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
102	98, 101	syl 17	. . . . . . . 8 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
103	102	eleq2d 2687	. . . . . . 7 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))}))
104		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘0) = ((𝑊 ++ ⟨“𝑣”⟩)‘0))
105	104	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋))
106		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)))
107	106, 104	neeq12d 2855	. . . . . . . . 9 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
108	105, 107	anbi12d 747	. . . . . . . 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))))
109	108	elrab 3363	. . . . . . 7 ⊢ ((𝑊 ++ ⟨“𝑣”⟩) ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))} ↔ ((𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))))
110	103, 109	syl6rbb 277	. . . . . 6 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
111	66, 92, 110	3bitrd 294	. . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
112	111	reubidva 3125	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → (∃!𝑣 ∈ 𝑉 ({( lastS ‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
113	42, 112	mpbid 222	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋))) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
114	113	ex 450	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
115	12, 114	sylbid 230	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃!wreu 2914 {crab 2916 Vcvv 3200 {cpr 4179 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ..^cfzo 12465 #chash 13117 Word cword 13291 lastS clsw 13292 ++ cconcat 13293 ⟨“cs1 13294 Vtxcvtx 25874 Edgcedg 25939 WWalksN cwwlksn 26718 ClWWalksN cclwwlksn 26876 FriendGraph cfrgr 27120

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-wwlks 26722 df-wwlksn 26723 df-clwwlks 26877 df-clwwlksn 26878 df-frgr 27121

This theorem is referenced by: numclwlk2lem2f1o 27238

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