numclwwlk2lem1 - Metamath Proof Explorer

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Theorem numclwwlk2lem1 30727

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 = (Vtx‘𝐺)
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
numclwwlk.h	⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})

**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) 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 30725	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
4	3	3adant1 1148	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
5	4	eleq2d 2849	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6		fveq1 6880	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
7	6	eqeq1d 2765	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
8		fveq2 6881	. . . . . 6 ⊢ (𝑤 = 𝑊 → (lastS‘𝑤) = (lastS‘𝑊))
9	8	neeq1d 3017	. . . . 5 ⊢ (𝑤 = 𝑊 → ((lastS‘𝑤) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ 𝑋))
10	7, 9	anbi12d 643	. . . 4 ⊢ (𝑤 = 𝑊 → (((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋) ↔ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)))
11	10	elrab 3650	. . 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	bitrdi 290	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))))
13		simpl1 1210	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝐺 ∈ FriendGraph )
14		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
15	1, 14	wwlknp 30192	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
16		peano2nn 12240	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
17	16	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
18		simpl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))
19	17, 18	jca 520	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1))))
20	19	ex 417	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))))
21	20	3adant3 1150	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))))
23		lswlgt0cl 14602	. . . . . . . . . . 11 ⊢ (((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1))) → (lastS‘𝑊) ∈ 𝑉)
24	22, 23	syl6 36	. . . . . . . . . 10 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → (lastS‘𝑊) ∈ 𝑉))
25	24	adantr 485	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑁 ∈ ℕ → (lastS‘𝑊) ∈ 𝑉))
26	25	com12 33	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ∈ 𝑉))
27	26	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ∈ 𝑉))
28	27	imp 411	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (lastS‘𝑊) ∈ 𝑉)
29		eleq1 2851	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
30	29	biimprd 251	. . . . . . . . . 10 ⊢ ((𝑊‘0) = 𝑋 → (𝑋 ∈ 𝑉 → (𝑊‘0) ∈ 𝑉))
31	30	ad2antrl 740	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑋 ∈ 𝑉 → (𝑊‘0) ∈ 𝑉))
32	31	com12 33	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
33	32	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
34	33	imp 411	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (𝑊‘0) ∈ 𝑉)
35		neeq2 3021	. . . . . . . . . 10 ⊢ (𝑋 = (𝑊‘0) → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
36	35	eqcoms 2771	. . . . . . . . 9 ⊢ ((𝑊‘0) = 𝑋 → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
37	36	biimpa 481	. . . . . . . 8 ⊢ (((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋) → (lastS‘𝑊) ≠ (𝑊‘0))
38	37	adantl 486	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ≠ (𝑊‘0))
39	38	adantl 486	. . . . . 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 1146	. . . . 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 30618	. . . . 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 66	. . . 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 487	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
44	43	ad2antlr 739	. . . . . . . . 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 489	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
46		nnnn0 12506	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
47	46	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
48	47	ad2antrr 738	. . . . . . . . 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 1146	. . . . . . . 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 30408	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣 ∈ 𝑉) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
51	50	3adant3 1150	. . . . . . . . 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)))
52	51	imp 411	. . . . . . . 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))
53	49, 52	sylan 591	. . . . . . 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))
54	1	wwlknbp 30191	. . . . . . . . . . 11 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉))
55	54	simp3d 1162	. . . . . . . . . 10 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ Word 𝑉)
56	55	ad2antrl 740	. . . . . . . . 9 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
57	56	ad2antrr 738	. . . . . . . 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)
58	45	adantr 485	. . . . . . . 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)
59		2z 12621	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
60		nn0pzuz 12924	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ_≥‘2))
61	46, 59, 60	sylancl 597	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ_≥‘2))
62	61	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 2) ∈ (ℤ_≥‘2))
63	62	ad3antrrr 742	. . . . . . . 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))
64		simpr 489	. . . . . . . 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))
65	1, 14	clwwlkext2edg 30407	. . . . . . . 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)))
66	57, 58, 63, 64, 65	syl31anc 1400	. . . . . . 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)))
67	53, 66	impbida 812	. . . . . 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)))
68	45, 1	eleqtrdi 2873	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (Vtx‘𝐺))
69	37	anim2i 628	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)))
70	69	ad2antlr 739	. . . . . . . . . . 11 ⊢ ((((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)))
71	70	simprd 500	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (lastS‘𝑊) ≠ (𝑊‘0))
72		numclwwlk2lem1lem 30693	. . . . . . . . . 10 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0)))
73	68, 44, 71, 72	syl3anc 1398	. . . . . . . . 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)))
74		eqeq2 2775	. . . . . . . . . . . . 13 ⊢ (𝑋 = (𝑊‘0) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
75	74	eqcoms 2771	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
76	75	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
77	76	ad2antlr 739	. . . . . . . . . 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)))
78	73	simpld 499	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0))
79	78	neeq2d 3018	. . . . . . . . . 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)))
80	77, 79	anbi12d 643	. . . . . . . . 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))))
81	73, 80	mpbird 260	. . . . . . . 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)))
82		nncn 12236	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
83		2cnd 12314	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
84	82, 83	pncand 11565	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 2) − 2) = 𝑁)
85	84	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑁 + 2) − 2) = 𝑁)
86	85	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → ((𝑁 + 2) − 2) = 𝑁)
87	86	fveq2d 6885	. . . . . . . . . 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))
88	87	neeq1d 3017	. . . . . . . . 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)))
89	88	anbi2d 641	. . . . . . . 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))))
90	81, 89	mpbird 260	. . . . . . 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)))
91	90	biantrud 540	. . . . . 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)))))
92	61	anim2i 628	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)))
93	92	3adant1 1148	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)))
94	93	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)))
95		numclwwlk.h	. . . . . . . . . 10 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
96	95	numclwwlkovh 30724	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
97	94, 96	syl 18	. . . . . . . 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))})
98	97	eleq2d 2849	. . . . . . 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))}))
99		fveq1 6880	. . . . . . . . . 10 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘0) = ((𝑊 ++ ⟨“𝑣”⟩)‘0))
100	99	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋))
101		fveq1 6880	. . . . . . . . . 10 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)))
102	101, 99	neeq12d 3019	. . . . . . . . 9 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
103	100, 102	anbi12d 643	. . . . . . . 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))))
104	103	elrab 3650	. . . . . . 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))))
105	98, 104	bitr2di 291	. . . . . 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))))
106	67, 91, 105	3bitrd 308	. . . . 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))))
107	106	reubidva 3383	. . . 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))))
108	42, 107	mpbid 235	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
109	108	ex 417	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
110	12, 109	sylbid 243	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 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃!wreu 3367 {crab 3416 Vcvv 3455 {cpr 4591 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 0cc0 11095 1c1 11096 + caddc 11098 − cmin 11436 ℕcn 12228 2c2 12290 ℕ₀cn0 12499 ℤcz 12586 ℤ_≥cuz 12857 ..^cfzo 13678 ♯chash 14362 Word cword 14546 lastSclsw 14595 ++ cconcat 14603 ⟨“cs1 14629 Vtxcvtx 29346 Edgcedg 29397 WWalksN cwwlksn 30175 ClWWalksN cclwwlkn 30375 ClWWalksNOncclwwlknon 30438 FriendGraph cfrgr 30609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-n0 12500 df-xnn0 12573 df-z 12587 df-uz 12858 df-rp 13012 df-fz 13531 df-fzo 13679 df-hash 14363 df-word 14547 df-lsw 14596 df-concat 14604 df-s1 14630 df-wwlks 30179 df-wwlksn 30180 df-clwwlk 30333 df-clwwlkn 30376 df-clwwlknon 30439 df-frgr 30610

This theorem is referenced by: numclwlk2lem2f1o 30730

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