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Theorem clwwlkf 30033

Description: Lemma 1 for clwwlkf1o 30037: 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 6881	. . . . 5 ⊢ (𝑤 = 𝑡 → (lastS‘𝑤) = (lastS‘𝑡))
2		fveq1 6880	. . . . 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 3679	. . 3 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑡) = (𝑡‘0)))
6		nnnn0 12513	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
7		iswwlksn 29825	. . . . . . . 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 29823	. . . . . . . . 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 12546	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
17	6, 16	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
18		nnre 12252	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
19	18	lep1d 12178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
20		elfz2nn0 13640	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (0...(𝑁 + 1)) ↔ (𝑁 ∈ ℕ₀ ∧ (𝑁 + 1) ∈ ℕ₀ ∧ 𝑁 ≤ (𝑁 + 1)))
21	6, 17, 19, 20	syl3anbrc 1344	. . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (0...(♯‘𝑡)) = (0...(𝑁 + 1)))
24	23	eleq2d 2821	. . . . . . . . . . . . . . . . 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 14706	. . . . . . . . . . . . 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 1187	. . . . . . . . . 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 14700	. . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑡) = (𝑁 + 1) → ((♯‘𝑡) − 1) = ((𝑁 + 1) − 1))
40	39	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (0..^((♯‘𝑡) − 1)) = (0..^((𝑁 + 1) − 1)))
41		nncn 12253	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
42		1cnd 11235	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
43	41, 42	pncand 11600	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
44	43	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (0..^((𝑁 + 1) − 1)) = (0..^𝑁))
45	40, 44	sylan9eqr 2793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (♯‘𝑡) = (𝑁 + 1)) → (0..^((♯‘𝑡) − 1)) = (0..^𝑁))
46	45	raleqdv 3309	. . . . . . . . . . . . . . . . . . . 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 12614	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
48		peano2zm 12640	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
49	47, 48	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
50	18	lem1d 12180	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ≤ 𝑁)
51		eluz2 12863	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) ↔ ((𝑁 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 − 1) ≤ 𝑁))
52	49, 47, 50, 51	syl3anbrc 1344	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
53		fzoss2 13709	. . . . . . . . . . . . . . . . . . . . . . . 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 4032	. . . . . . . . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4722	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝑡 prefix 𝑁)) = 𝑁 → ((♯‘(𝑡 prefix 𝑁)) − 1) = (𝑁 − 1))
92	91	oveq2d 7426	. . . . . . . . . . . . 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 3309	. . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . 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 12705	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℤ)
99		fznn 13614	. . . . . . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑡) = (𝑁 + 1) → (1...(♯‘𝑡)) = (1...(𝑁 + 1)))
104	103	eleq2d 2821	. . . . . . . . . . . . . . . . . . 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 14718	. . . . . . . . . . . . . 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 14715	. . . . . . . . . . . . . . 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 4722	. . . . . . . . . . . 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 14587	. . . . . . . . . . . . . . . . . 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 2793	. . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 4721	. . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((♯‘𝑡) − 1) = 𝑁)
130	129	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (0..^((♯‘𝑡) − 1)) = (0..^𝑁))
131	130	raleqdv 3309	. . . . . . . . . . . . . . . . . . . 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 13779	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (0..^𝑁))
133		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = (𝑁 − 1) → (𝑡‘𝑖) = (𝑡‘(𝑁 − 1)))
134		fvoveq1 7433	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = (𝑁 − 1) → (𝑡‘(𝑖 + 1)) = (𝑡‘((𝑁 − 1) + 1)))
135	133, 134	preq12d 4722	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = (𝑁 − 1) → {(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} = {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))})
136	135	eleq1d 2820	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = (𝑁 − 1) → ({(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ (Edg‘𝐺)))
137	136	rspcva 3604	. . . . . . . . . . . . . . . . . . . . . . . 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 11603	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁)
140	139	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → (𝑡‘((𝑁 − 1) + 1)) = (𝑡‘𝑁))
141	140	preq2d 4721	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} = {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)})
142	141	eleq1d 2820	. . . . . . . . . . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . 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 2835	. . . . . . . . . . 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 30022	. . . . 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 7109	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 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 {crab 3420 ⊆ wss 3931 ∅c0 4313 {cpr 4608 class class class wbr 5124 ↦ cmpt 5206 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 0cc0 11134 1c1 11135 + caddc 11137 ≤ cle 11275 − cmin 11471 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529 ..^cfzo 13676 ♯chash 14353 Word cword 14536 lastSclsw 14585 prefix cpfx 14693 Vtxcvtx 28980 Edgcedg 29031 WWalkscwwlks 29812 WWalksN cwwlksn 29813 ClWWalksN cclwwlkn 30010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 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 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-hash 14354 df-word 14537 df-lsw 14586 df-substr 14664 df-pfx 14694 df-wwlks 29817 df-wwlksn 29818 df-clwwlk 29968 df-clwwlkn 30011

This theorem is referenced by: clwwlkf1 30035 clwwlkfo 30036

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