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Theorem numclwwlk2lem1 30405

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 30403	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
4	3	3adant1 1129	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
5	4	eleq2d 2825	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6		fveq1 6906	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
7	6	eqeq1d 2737	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
8		fveq2 6907	. . . . . 6 ⊢ (𝑤 = 𝑊 → (lastS‘𝑤) = (lastS‘𝑊))
9	8	neeq1d 2998	. . . . 5 ⊢ (𝑤 = 𝑊 → ((lastS‘𝑤) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ 𝑋))
10	7, 9	anbi12d 632	. . . 4 ⊢ (𝑤 = 𝑊 → (((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋) ↔ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)))
11	10	elrab 3695	. . 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 287	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))))
13		simpl1 1190	. . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝐺 ∈ FriendGraph )
14		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
15	1, 14	wwlknp 29873	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
16		peano2nn 12276	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
17	16	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
18		simpl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))
19	17, 18	jca 511	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1))))
20	19	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))))
21	20	3adant3 1131	. . . . . . . . . . . 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 14604	. . . . . . . . . . 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 480	. . . . . . . . 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 1134	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ∈ 𝑉))
28	27	imp 406	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (lastS‘𝑊) ∈ 𝑉)
29		eleq1 2827	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
30	29	biimprd 248	. . . . . . . . . 10 ⊢ ((𝑊‘0) = 𝑋 → (𝑋 ∈ 𝑉 → (𝑊‘0) ∈ 𝑉))
31	30	ad2antrl 728	. . . . . . . . 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 1133	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
34	33	imp 406	. . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (𝑊‘0) ∈ 𝑉)
35		neeq2 3002	. . . . . . . . . 10 ⊢ (𝑋 = (𝑊‘0) → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
36	35	eqcoms 2743	. . . . . . . . 9 ⊢ ((𝑊‘0) = 𝑋 → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
37	36	biimpa 476	. . . . . . . 8 ⊢ (((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋) → (lastS‘𝑊) ≠ (𝑊‘0))
38	37	adantl 481	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ≠ (𝑊‘0))
39	38	adantl 481	. . . . . 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 1127	. . . . 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 30296	. . . . 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 482	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
44	43	ad2antlr 727	. . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
46		nnnn0 12531	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
47	46	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
48	47	ad2antrr 726	. . . . . . . . 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 1127	. . . . . . . 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 30086	. . . . . . . . . 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 1131	. . . . . . . . 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 406	. . . . . . . 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 580	. . . . . . 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 29872	. . . . . . . . . . 11 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉))
55	54	simp3d 1143	. . . . . . . . . 10 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ Word 𝑉)
56	55	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
57	56	ad2antrr 726	. . . . . . . 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 480	. . . . . . . 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 12647	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
60		nn0pzuz 12945	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ₀ ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ_≥‘2))
61	46, 59, 60	sylancl 586	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ_≥‘2))
62	61	3ad2ant3 1134	. . . . . . . . 9 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑁 + 2) ∈ (ℤ_≥‘2))
63	62	ad3antrrr 730	. . . . . . . 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 484	. . . . . . . 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 30085	. . . . . . . 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 1372	. . . . . . 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 801	. . . . . 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 2849	. . . . . . . . . 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 617	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)))
70	69	ad2antlr 727	. . . . . . . . . . 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 495	. . . . . . . . . 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 30371	. . . . . . . . . 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 1370	. . . . . . . . 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 2747	. . . . . . . . . . . . 13 ⊢ (𝑋 = (𝑊‘0) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
75	74	eqcoms 2743	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
76	75	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
77	76	ad2antlr 727	. . . . . . . . . 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 494	. . . . . . . . . . 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 2999	. . . . . . . . . 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 632	. . . . . . . . 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 257	. . . . . . . 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 12272	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
83		2cnd 12342	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
84	82, 83	pncand 11619	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 2) − 2) = 𝑁)
85	84	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑁 + 2) − 2) = 𝑁)
86	85	ad2antrr 726	. . . . . . . . . . 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 6911	. . . . . . . . . 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 2998	. . . . . . . . 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 630	. . . . . . . 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 257	. . . . . . 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 531	. . . . . 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 617	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)))
93	92	3adant1 1129	. . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)))
94	93	ad2antrr 726	. . . . . . . . 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 30402	. . . . . . . . 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 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))})
98	97	eleq2d 2825	. . . . . . 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 6906	. . . . . . . . . 10 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘0) = ((𝑊 ++ ⟨“𝑣”⟩)‘0))
100	99	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋))
101		fveq1 6906	. . . . . . . . . 10 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)))
102	101, 99	neeq12d 3000	. . . . . . . . 9 ⊢ (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
103	100, 102	anbi12d 632	. . . . . . . 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 3695	. . . . . . 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 288	. . . . . 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 305	. . . . 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 3394	. . . 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 232	. . 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 412	. 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 240	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃!wreu 3376 {crab 3433 Vcvv 3478 {cpr 4633 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 0cc0 11153 1c1 11154 + caddc 11156 − cmin 11490 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ..^cfzo 13691 ♯chash 14366 Word cword 14549 lastSclsw 14597 ++ cconcat 14605 ⟨“cs1 14630 Vtxcvtx 29028 Edgcedg 29079 WWalksN cwwlksn 29856 ClWWalksN cclwwlkn 30053 ClWWalksNOncclwwlknon 30116 FriendGraph cfrgr 30287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-wwlks 29860 df-wwlksn 29861 df-clwwlk 30011 df-clwwlkn 30054 df-clwwlknon 30117 df-frgr 30288

This theorem is referenced by: numclwlk2lem2f1o 30408

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