clwwlkf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > clwwlkf		Structured version Visualization version GIF version

Theorem clwwlkf 30067

Description: Lemma 1 for clwwlkf1o 30071: F is a function. (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)

**Hypotheses**
Ref	Expression
clwwlkf1o.d	⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
clwwlkf1o.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))

**Assertion**
Ref	Expression
clwwlkf	⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁 𝑡,𝐷 𝑡,𝐺,𝑤 𝑡,𝑁 Allowed substitution hints: 𝐷(𝑤) 𝐹(𝑤,𝑡)

Proof of Theorem clwwlkf Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6905	. . . . 5 ⊢ (𝑤 = 𝑡 → (lastS‘𝑤) = (lastS‘𝑡))
2		fveq1 6904	. . . . 5 ⊢ (𝑤 = 𝑡 → (𝑤‘0) = (𝑡‘0))
3	1, 2	eqeq12d 2752	. . . 4 ⊢ (𝑤 = 𝑡 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑡) = (𝑡‘0)))
4		clwwlkf1o.d	. . . 4 ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
5	3, 4	elrab2 3694	. . 3 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑡) = (𝑡‘0)))
6		nnnn0 12535	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
7		iswwlksn 29859	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (𝑡 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑡 ∈ (WWalks‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1))))
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑡 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑡 ∈ (WWalks‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1))))
9		eqid 2736	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10		eqid 2736	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
11	9, 10	iswwlks 29857	. . . . . . . . 9 ⊢ (𝑡 ∈ (WWalks‘𝐺) ↔ (𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
12	11	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑡 ∈ (WWalks‘𝐺) ↔ (𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
13	12	anbi1d 631	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑡 ∈ (WWalks‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) ↔ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))))
14	8, 13	bitrd 279	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑡 ∈ (𝑁 WWalksN 𝐺) ↔ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))))
15		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑡 ∈ Word (Vtx‘𝐺))
16		peano2nn0 12568	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
17	6, 16	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
18		nnre 12274	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
19	18	lep1d 12200	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
20		elfz2nn0 13659	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (0...(𝑁 + 1)) ↔ (𝑁 ∈ ℕ₀ ∧ (𝑁 + 1) ∈ ℕ₀ ∧ 𝑁 ≤ (𝑁 + 1)))
21	6, 17, 19, 20	syl3anbrc 1343	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (0...(𝑁 + 1)))
22	21	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (0...(𝑁 + 1)))
23		oveq2 7440	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (0...(♯‘𝑡)) = (0...(𝑁 + 1)))
24	23	eleq2d 2826	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (𝑁 ∈ (0...(♯‘𝑡)) ↔ 𝑁 ∈ (0...(𝑁 + 1))))
25	24	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ (0...(♯‘𝑡)) ↔ 𝑁 ∈ (0...(𝑁 + 1))))
26	25	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ (0...(♯‘𝑡)) ↔ 𝑁 ∈ (0...(𝑁 + 1))))
27	22, 26	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (0...(♯‘𝑡)))
28	15, 27	jca 511	. . . . . . . . . . . . 13 ⊢ (((𝑡 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑡))))
29		pfxlen 14722	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑡))) → (♯‘(𝑡 prefix 𝑁)) = 𝑁)
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (♯‘(𝑡 prefix 𝑁)) = 𝑁)
31	30	ex 412	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (♯‘(𝑡 prefix 𝑁)) = 𝑁))
32	31	3ad2antl2 1186	. . . . . . . . . 10 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (♯‘(𝑡 prefix 𝑁)) = 𝑁))
33	32	impcom 407	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → (♯‘(𝑡 prefix 𝑁)) = 𝑁)
34	33	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → (♯‘(𝑡 prefix 𝑁)) = 𝑁)
35		pfxcl 14716	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ Word (Vtx‘𝐺) → (𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺))
36	35	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺))
37	36	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → (𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺))
38	37	ad2antrl 728	. . . . . . . . . 10 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → (𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺))
39		oveq1 7439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑡) = (𝑁 + 1) → ((♯‘𝑡) − 1) = ((𝑁 + 1) − 1))
40	39	oveq2d 7448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (0..^((♯‘𝑡) − 1)) = (0..^((𝑁 + 1) − 1)))
41		nncn 12275	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
42		1cnd 11257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
43	41, 42	pncand 11622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
44	43	oveq2d 7448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (0..^((𝑁 + 1) − 1)) = (0..^𝑁))
45	40, 44	sylan9eqr 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (0..^((♯‘𝑡) − 1)) = (0..^𝑁))
46	45	raleqdv 3325	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
47		nnz 12636	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
48		peano2zm 12662	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
49	47, 48	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
50	18	lem1d 12202	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ≤ 𝑁)
51		eluz2 12885	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) ↔ ((𝑁 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 − 1) ≤ 𝑁))
52	49, 47, 50, 51	syl3anbrc 1343	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
53		fzoss2 13728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
55	54	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
56		ssralv 4051	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0..^(𝑁 − 1)) ⊆ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
57	55, 56	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
58		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑡 ∈ Word (Vtx‘𝐺))
59	21	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → 𝑁 ∈ (0...(𝑁 + 1)))
60	24	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ (0...(♯‘𝑡)) ↔ 𝑁 ∈ (0...(𝑁 + 1))))
61	59, 60	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → 𝑁 ∈ (0...(♯‘𝑡)))
62	61	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑁 ∈ (0...(♯‘𝑡)))
63	54	sseld 3981	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ ℕ → (𝑖 ∈ (0..^(𝑁 − 1)) → 𝑖 ∈ (0..^𝑁)))
64	63	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) → (𝑖 ∈ (0..^(𝑁 − 1)) → 𝑖 ∈ (0..^𝑁)))
65	64	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑖 ∈ (0..^𝑁))
66		pfxfv 14721	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑡)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑡 prefix 𝑁)‘𝑖) = (𝑡‘𝑖))
67	66	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑡)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑡‘𝑖) = ((𝑡 prefix 𝑁)‘𝑖))
68	58, 62, 65, 67	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑡‘𝑖) = ((𝑡 prefix 𝑁)‘𝑖))
69	47	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) → 𝑁 ∈ ℤ)
70		elfzom1elp1fzo 13772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
71	69, 70	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
72		pfxfv 14721	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑡)) ∧ (𝑖 + 1) ∈ (0..^𝑁)) → ((𝑡 prefix 𝑁)‘(𝑖 + 1)) = (𝑡‘(𝑖 + 1)))
73	72	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑡)) ∧ (𝑖 + 1) ∈ (0..^𝑁)) → (𝑡‘(𝑖 + 1)) = ((𝑡 prefix 𝑁)‘(𝑖 + 1)))
74	58, 62, 71, 73	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑡‘(𝑖 + 1)) = ((𝑡 prefix 𝑁)‘(𝑖 + 1)))
75	68, 74	preq12d 4740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → {(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} = {((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))})
76	75	eleq1d 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → ({(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
77	76	ralbidva 3175	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
78	77	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word (Vtx‘𝐺)) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
79	78	ex 412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (𝑡 ∈ Word (Vtx‘𝐺) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
80	79	com23 86	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝑡 ∈ Word (Vtx‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
81	57, 80	syld 47	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝑡 ∈ Word (Vtx‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
82	46, 81	sylbid 240	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝑡 ∈ Word (Vtx‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
83	82	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → ((♯‘𝑡) = (𝑁 + 1) → (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝑡 ∈ Word (Vtx‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
84	83	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ((♯‘𝑡) = (𝑁 + 1) → (𝑡 ∈ Word (Vtx‘𝐺) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
85	84	com14 96	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ Word (Vtx‘𝐺) → (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ((♯‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))))
86	85	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((♯‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
87	86	3adant1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((♯‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
88	87	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
89	88	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
90	89	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
91		oveq1 7439	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝑡 prefix 𝑁)) = 𝑁 → ((♯‘(𝑡 prefix 𝑁)) − 1) = (𝑁 − 1))
92	91	oveq2d 7448	. . . . . . . . . . . . 13 ⊢ ((♯‘(𝑡 prefix 𝑁)) = 𝑁 → (0..^((♯‘(𝑡 prefix 𝑁)) − 1)) = (0..^(𝑁 − 1)))
93	92	adantr 480	. . . . . . . . . . . 12 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → (0..^((♯‘(𝑡 prefix 𝑁)) − 1)) = (0..^(𝑁 − 1)))
94	93	raleqdv 3325	. . . . . . . . . . 11 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → (∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
95	90, 94	mpbird 257	. . . . . . . . . 10 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
96		simprl2 1219	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → 𝑡 ∈ Word (Vtx‘𝐺))
97	19	ancli 548	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℕ ∧ 𝑁 ≤ (𝑁 + 1)))
98	47	peano2zd 12727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℤ)
99		fznn 13633	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 + 1) ∈ ℤ → (𝑁 ∈ (1...(𝑁 + 1)) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ (𝑁 + 1))))
100	98, 99	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ (1...(𝑁 + 1)) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ (𝑁 + 1))))
101	97, 100	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
102	101	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → 𝑁 ∈ (1...(𝑁 + 1)))
103		oveq2 7440	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (1...(♯‘𝑡)) = (1...(𝑁 + 1)))
104	103	eleq2d 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (𝑁 ∈ (1...(♯‘𝑡)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
105	104	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ (1...(♯‘𝑡)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
106	105	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → (𝑁 ∈ (1...(♯‘𝑡)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
107	102, 106	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → 𝑁 ∈ (1...(♯‘𝑡)))
108	96, 107	jca 511	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → (𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑡))))
109	108	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → (𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑡))))
110		pfxfvlsw 14734	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑡))) → (lastS‘(𝑡 prefix 𝑁)) = (𝑡‘(𝑁 − 1)))
111	109, 110	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → (lastS‘(𝑡 prefix 𝑁)) = (𝑡‘(𝑁 − 1)))
112		pfxfv0 14731	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ (1...(♯‘𝑡))) → ((𝑡 prefix 𝑁)‘0) = (𝑡‘0))
113	108, 112	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → ((𝑡 prefix 𝑁)‘0) = (𝑡‘0))
114	113	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → ((𝑡 prefix 𝑁)‘0) = (𝑡‘0))
115	111, 114	preq12d 4740	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} = {(𝑡‘(𝑁 − 1)), (𝑡‘0)})
116		eqcom 2743	. . . . . . . . . . . . . . . . 17 ⊢ ((lastS‘𝑡) = (𝑡‘0) ↔ (𝑡‘0) = (lastS‘𝑡))
117	116	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ ((lastS‘𝑡) = (𝑡‘0) → (𝑡‘0) = (lastS‘𝑡))
118	117	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → (𝑡‘0) = (lastS‘𝑡))
119		lsw 14603	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ Word (Vtx‘𝐺) → (lastS‘𝑡) = (𝑡‘((♯‘𝑡) − 1)))
120	119	3ad2ant2 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (lastS‘𝑡) = (𝑡‘((♯‘𝑡) − 1)))
121	120	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → (lastS‘𝑡) = (𝑡‘((♯‘𝑡) − 1)))
122	121	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → (lastS‘𝑡) = (𝑡‘((♯‘𝑡) − 1)))
123	39	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1)) → ((♯‘𝑡) − 1) = ((𝑁 + 1) − 1))
124	123, 43	sylan9eqr 2798	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → ((♯‘𝑡) − 1) = 𝑁)
125	124	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → ((♯‘𝑡) − 1) = 𝑁)
126	125	fveq2d 6909	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → (𝑡‘((♯‘𝑡) − 1)) = (𝑡‘𝑁))
127	118, 122, 126	3eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → (𝑡‘0) = (𝑡‘𝑁))
128	127	preq2d 4739	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → {(𝑡‘(𝑁 − 1)), (𝑡‘0)} = {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)})
129	39, 43	sylan9eq 2796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((♯‘𝑡) − 1) = 𝑁)
130	129	oveq2d 7448	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (0..^((♯‘𝑡) − 1)) = (0..^𝑁))
131	130	raleqdv 3325	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
132		fzo0end 13798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (0..^𝑁))
133		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = (𝑁 − 1) → (𝑡‘𝑖) = (𝑡‘(𝑁 − 1)))
134		fvoveq1 7455	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = (𝑁 − 1) → (𝑡‘(𝑖 + 1)) = (𝑡‘((𝑁 − 1) + 1)))
135	133, 134	preq12d 4740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = (𝑁 − 1) → {(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} = {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))})
136	135	eleq1d 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = (𝑁 − 1) → ({(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ (Edg‘𝐺)))
137	136	rspcva 3619	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 − 1) ∈ (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ (Edg‘𝐺))
138	132, 137	sylan 580	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ (Edg‘𝐺))
139	41, 42	npcand 11625	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁)
140	139	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → (𝑡‘((𝑁 − 1) + 1)) = (𝑡‘𝑁))
141	140	preq2d 4739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} = {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)})
142	141	eleq1d 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → ({(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ (Edg‘𝐺) ↔ {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺)))
143	142	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → ({(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ (Edg‘𝐺) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺)))
144	143	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ({(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ (Edg‘𝐺) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺)))
145	138, 144	mpd 15	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺))
146	145	ex 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺)))
147	146	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺)))
148	131, 147	sylbid 240	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺)))
149	148	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺))))
150	149	com3r 87	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ((♯‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺))))
151	150	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((♯‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺))))
152	151	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺)))
153	152	impcom 407	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺))
154	153	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ (Edg‘𝐺))
155	128, 154	eqeltrd 2840	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → {(𝑡‘(𝑁 − 1)), (𝑡‘0)} ∈ (Edg‘𝐺))
156	115, 155	eqeltrd 2840	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺))
157	156	adantl 481	. . . . . . . . . 10 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺))
158	38, 95, 157	3jca 1128	. . . . . . . . 9 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → ((𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)))
159		simpl 482	. . . . . . . . 9 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → (♯‘(𝑡 prefix 𝑁)) = 𝑁)
160	158, 159	jca 511	. . . . . . . 8 ⊢ (((♯‘(𝑡 prefix 𝑁)) = 𝑁 ∧ ((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0))) → (((𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘(𝑡 prefix 𝑁)) = 𝑁))
161	34, 160	mpancom 688	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1))) ∧ (lastS‘𝑡) = (𝑡‘0)) → (((𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘(𝑡 prefix 𝑁)) = 𝑁))
162	161	exp31 419	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (♯‘𝑡) = (𝑁 + 1)) → ((lastS‘𝑡) = (𝑡‘0) → (((𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘(𝑡 prefix 𝑁)) = 𝑁))))
163	14, 162	sylbid 240	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑡 ∈ (𝑁 WWalksN 𝐺) → ((lastS‘𝑡) = (𝑡‘0) → (((𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘(𝑡 prefix 𝑁)) = 𝑁))))
164	163	imp32 418	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑡 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑡) = (𝑡‘0))) → (((𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘(𝑡 prefix 𝑁)) = 𝑁))
165	9, 10	isclwwlknx 30056	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑡 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺) ↔ (((𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘(𝑡 prefix 𝑁)) = 𝑁)))
166	165	adantr 480	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑡 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑡) = (𝑡‘0))) → ((𝑡 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺) ↔ (((𝑡 prefix 𝑁) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(𝑡 prefix 𝑁)) − 1)){((𝑡 prefix 𝑁)‘𝑖), ((𝑡 prefix 𝑁)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘(𝑡 prefix 𝑁)), ((𝑡 prefix 𝑁)‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘(𝑡 prefix 𝑁)) = 𝑁)))
167	164, 166	mpbird 257	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑡 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑡) = (𝑡‘0))) → (𝑡 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))
168	5, 167	sylan2b 594	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑡 ∈ 𝐷) → (𝑡 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))
169		clwwlkf1o.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))
170	168, 169	fmptd 7133	1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 {crab 3435 ⊆ wss 3950 ∅c0 4332 {cpr 4627 class class class wbr 5142 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 + caddc 11159 ≤ cle 11297 − cmin 11493 ℕcn 12267 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12879 ...cfz 13548 ..^cfzo 13695 ♯chash 14370 Word cword 14553 lastSclsw 14601 prefix cpfx 14709 Vtxcvtx 29014 Edgcedg 29065 WWalkscwwlks 29846 WWalksN cwwlksn 29847 ClWWalksN cclwwlkn 30044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-lsw 14602 df-substr 14680 df-pfx 14710 df-wwlks 29851 df-wwlksn 29852 df-clwwlk 30002 df-clwwlkn 30045

This theorem is referenced by: clwwlkf1 30069 clwwlkfo 30070

W3C validator