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Theorem clwwlkf 26322

Description: Lemma 1 for clwwlkbij 26327: F is a function. (Contributed by Alexander van der Vekens, 27-Sep-2018.)

**Hypotheses**
Ref	Expression
clwwlkbij.d	⊢ 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
clwwlkbij.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))

**Assertion**
Ref	Expression
clwwlkf	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → 𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁))

Distinct variable groups: 𝑤,𝐸 𝑤,𝑁 𝑤,𝑉 𝑡,𝐷 𝑡,𝐸,𝑤 𝑡,𝑁 𝑡,𝑉 𝑡,𝑋 𝑡,𝑌 Allowed substitution hints: 𝐷(𝑤) 𝐹(𝑤,𝑡) 𝑋(𝑤) 𝑌(𝑤)

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

Step	Hyp	Ref	Expression
1		fveq2 6103	. . . . 5 ⊢ (𝑤 = 𝑡 → ( lastS ‘𝑤) = ( lastS ‘𝑡))
2		fveq1 6102	. . . . 5 ⊢ (𝑤 = 𝑡 → (𝑤‘0) = (𝑡‘0))
3	1, 2	eqeq12d 2625	. . . 4 ⊢ (𝑤 = 𝑡 → (( lastS ‘𝑤) = (𝑤‘0) ↔ ( lastS ‘𝑡) = (𝑡‘0)))
4		clwwlkbij.d	. . . 4 ⊢ 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
5	3, 4	elrab2 3333	. . 3 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑡) = (𝑡‘0)))
6		nnnn0 11176	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
7		iswwlkn 26212	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ₀) → (𝑡 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑡 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))))
8	6, 7	syl3an3 1353	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (𝑡 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑡 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))))
9		iswwlk 26211	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑡 ∈ (𝑉 WWalks 𝐸) ↔ (𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸)))
10	9	3adant3 1074	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (𝑡 ∈ (𝑉 WWalks 𝐸) ↔ (𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸)))
11	10	anbi1d 737	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑡 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) ↔ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))))
12	8, 11	bitrd 267	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (𝑡 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))))
13		simpll 786	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡 ∈ Word 𝑉 ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑡 ∈ Word 𝑉)
14		peano2nn0 11210	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
15	6, 14	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
16		nnre 10904	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
17	16	lep1d 10834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
18		elfz2nn0 12300	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (0...(𝑁 + 1)) ↔ (𝑁 ∈ ℕ₀ ∧ (𝑁 + 1) ∈ ℕ₀ ∧ 𝑁 ≤ (𝑁 + 1)))
19	6, 15, 17, 18	syl3anbrc 1239	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (0...(𝑁 + 1)))
20	19	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡 ∈ Word 𝑉 ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (0...(𝑁 + 1)))
21		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑡) = (𝑁 + 1) → (0...(#‘𝑡)) = (0...(𝑁 + 1)))
22	21	eleq2d 2673	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑡) = (𝑁 + 1) → (𝑁 ∈ (0...(#‘𝑡)) ↔ 𝑁 ∈ (0...(𝑁 + 1))))
23	22	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ∈ Word 𝑉 ∧ (#‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ (0...(#‘𝑡)) ↔ 𝑁 ∈ (0...(𝑁 + 1))))
24	23	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡 ∈ Word 𝑉 ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ (0...(#‘𝑡)) ↔ 𝑁 ∈ (0...(𝑁 + 1))))
25	20, 24	mpbird 246	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡 ∈ Word 𝑉 ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (0...(#‘𝑡)))
26	13, 25	jca 553	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ Word 𝑉 ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑡))))
27		swrd0len 13274	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑡))) → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)
28	26, 27	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ Word 𝑉 ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)
29	28	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ Word 𝑉 ∧ (#‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))
30	29	3ad2antl2 1217	. . . . . . . . . . . 12 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))
31	30	com12 32	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))
32	31	3ad2ant3 1077	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))
33	32	imp 444	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)
34	33	adantr 480	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)
35		swrdcl 13271	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ Word 𝑉 → (𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉)
36	35	3ad2ant2 1076	. . . . . . . . . . . 12 ⊢ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉)
37	36	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → (𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉)
38	37	ad2antrl 760	. . . . . . . . . 10 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → (𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉)
39		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑡) = (𝑁 + 1) → ((#‘𝑡) − 1) = ((𝑁 + 1) − 1))
40	39	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑡) = (𝑁 + 1) → (0..^((#‘𝑡) − 1)) = (0..^((𝑁 + 1) − 1)))
41		nncn 10905	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
42		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
43	41, 42	pncand 10272	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
44	43	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (0..^((𝑁 + 1) − 1)) = (0..^𝑁))
45	40, 44	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (0..^((#‘𝑡) − 1)) = (0..^𝑁))
46	45	raleqdv 3121	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸))
47		nnz 11276	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
48		peano2zm 11297	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
49	47, 48	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℤ)
50	16	lem1d 10836	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ≤ 𝑁)
51		eluz2 11569	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) ↔ ((𝑁 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 − 1) ≤ 𝑁))
52	49, 47, 50, 51	syl3anbrc 1239	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
53		fzoss2 12365	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
55	54	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
56		ssralv 3629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0..^(𝑁 − 1)) ⊆ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸))
57	55, 56	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸))
58		simplr 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑡 ∈ Word 𝑉)
59	19	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → 𝑁 ∈ (0...(𝑁 + 1)))
60	22	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ (0...(#‘𝑡)) ↔ 𝑁 ∈ (0...(𝑁 + 1))))
61	59, 60	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → 𝑁 ∈ (0...(#‘𝑡)))
62	61	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑁 ∈ (0...(#‘𝑡)))
63	54	sseld 3567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑁 ∈ ℕ → (𝑖 ∈ (0..^(𝑁 − 1)) → 𝑖 ∈ (0..^𝑁)))
64	63	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) → (𝑖 ∈ (0..^(𝑁 − 1)) → 𝑖 ∈ (0..^𝑁)))
65	64	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → 𝑖 ∈ (0..^𝑁))
66		swrd0fv 13291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑡)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑡 substr ⟨0, 𝑁⟩)‘𝑖) = (𝑡‘𝑖))
67	66	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑡)) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑡‘𝑖) = ((𝑡 substr ⟨0, 𝑁⟩)‘𝑖))
68	58, 62, 65, 67	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑡‘𝑖) = ((𝑡 substr ⟨0, 𝑁⟩)‘𝑖))
69	47	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) → 𝑁 ∈ ℤ)
70		elfzom1elp1fzo 12402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
71	69, 70	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑖 + 1) ∈ (0..^𝑁))
72		swrd0fv 13291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑡)) ∧ (𝑖 + 1) ∈ (0..^𝑁)) → ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1)) = (𝑡‘(𝑖 + 1)))
73	72	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑡)) ∧ (𝑖 + 1) ∈ (0..^𝑁)) → (𝑡‘(𝑖 + 1)) = ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1)))
74	58, 62, 71, 73	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → (𝑡‘(𝑖 + 1)) = ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1)))
75	68, 74	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → {(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} = {((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))})
76	75	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) ∧ 𝑖 ∈ (0..^(𝑁 − 1))) → ({(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
77	76	ralbidva 2968	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
78	77	biimpd 218	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) ∧ 𝑡 ∈ Word 𝑉) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
79	78	ex 449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (𝑡 ∈ Word 𝑉 → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
80	79	com23 84	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑡 ∈ Word 𝑉 → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
81	57, 80	syld 46	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑡 ∈ Word 𝑉 → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
82	46, 81	sylbid 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (#‘𝑡) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑡 ∈ Word 𝑉 → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
83	82	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → ((#‘𝑡) = (𝑁 + 1) → (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑡 ∈ Word 𝑉 → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))))
84	83	com23 84	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → ((#‘𝑡) = (𝑁 + 1) → (𝑡 ∈ Word 𝑉 → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))))
85	84	com14 94	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ Word 𝑉 → (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → ((#‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))))
86	85	imp 444	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → ((#‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
87	86	3adant1 1072	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → ((#‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)))
88	87	imp 444	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
89	88	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
90	89	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
91	90	imp 444	. . . . . . . . . . . 12 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
92	91	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
93		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ ((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 → ((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1) = (𝑁 − 1))
94	93	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ ((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 → (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)) = (0..^(𝑁 − 1)))
95	94	adantr 480	. . . . . . . . . . . 12 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)) = (0..^(𝑁 − 1)))
96	95	raleqdv 3121	. . . . . . . . . . 11 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → (∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^(𝑁 − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸))
97	92, 96	mpbird 246	. . . . . . . . . 10 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸)
98		simprl2 1100	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → 𝑡 ∈ Word 𝑉)
99	17	ancli 572	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℕ ∧ 𝑁 ≤ (𝑁 + 1)))
100	47	peano2zd 11361	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℤ)
101		fznn 12278	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 + 1) ∈ ℤ → (𝑁 ∈ (1...(𝑁 + 1)) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ (𝑁 + 1))))
102	100, 101	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ (1...(𝑁 + 1)) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ (𝑁 + 1))))
103	99, 102	mpbird 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
104	103	3ad2ant3 1077	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
105	104	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → 𝑁 ∈ (1...(𝑁 + 1)))
106		oveq2 6557	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑡) = (𝑁 + 1) → (1...(#‘𝑡)) = (1...(𝑁 + 1)))
107	106	eleq2d 2673	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑡) = (𝑁 + 1) → (𝑁 ∈ (1...(#‘𝑡)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
108	107	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ (1...(#‘𝑡)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
109	108	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → (𝑁 ∈ (1...(#‘𝑡)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
110	105, 109	mpbird 246	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → 𝑁 ∈ (1...(#‘𝑡)))
111	98, 110	jca 553	. . . . . . . . . . . . . . 15 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → (𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑡))))
112	111	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → (𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑡))))
113		swrd0fvlsw 13295	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑡))) → ( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)) = (𝑡‘(𝑁 − 1)))
114	112, 113	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → ( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)) = (𝑡‘(𝑁 − 1)))
115		swrd0fv0 13292	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Word 𝑉 ∧ 𝑁 ∈ (1...(#‘𝑡))) → ((𝑡 substr ⟨0, 𝑁⟩)‘0) = (𝑡‘0))
116	111, 115	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → ((𝑡 substr ⟨0, 𝑁⟩)‘0) = (𝑡‘0))
117	116	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → ((𝑡 substr ⟨0, 𝑁⟩)‘0) = (𝑡‘0))
118	114, 117	preq12d 4220	. . . . . . . . . . . 12 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} = {(𝑡‘(𝑁 − 1)), (𝑡‘0)})
119		eqcom 2617	. . . . . . . . . . . . . . . . 17 ⊢ (( lastS ‘𝑡) = (𝑡‘0) ↔ (𝑡‘0) = ( lastS ‘𝑡))
120	119	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ (( lastS ‘𝑡) = (𝑡‘0) → (𝑡‘0) = ( lastS ‘𝑡))
121	120	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → (𝑡‘0) = ( lastS ‘𝑡))
122		lsw 13204	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ Word 𝑉 → ( lastS ‘𝑡) = (𝑡‘((#‘𝑡) − 1)))
123	122	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → ( lastS ‘𝑡) = (𝑡‘((#‘𝑡) − 1)))
124	123	ad2antrl 760	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → ( lastS ‘𝑡) = (𝑡‘((#‘𝑡) − 1)))
125	124	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → ( lastS ‘𝑡) = (𝑡‘((#‘𝑡) − 1)))
126	39	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → ((#‘𝑡) − 1) = ((𝑁 + 1) − 1))
127	43	3ad2ant3 1077	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) − 1) = 𝑁)
128	126, 127	sylan9eqr 2666	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → ((#‘𝑡) − 1) = 𝑁)
129	128	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → ((#‘𝑡) − 1) = 𝑁)
130	129	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → (𝑡‘((#‘𝑡) − 1)) = (𝑡‘𝑁))
131	121, 125, 130	3eqtrd 2648	. . . . . . . . . . . . . 14 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → (𝑡‘0) = (𝑡‘𝑁))
132	131	preq2d 4219	. . . . . . . . . . . . 13 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → {(𝑡‘(𝑁 − 1)), (𝑡‘0)} = {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)})
133	39, 43	sylan9eq 2664	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((#‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((#‘𝑡) − 1) = 𝑁)
134	133	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((#‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (0..^((#‘𝑡) − 1)) = (0..^𝑁))
135	134	raleqdv 3121	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((#‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸))
136		fzo0end 12426	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (0..^𝑁))
137		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 = (𝑁 − 1) → (𝑡‘𝑖) = (𝑡‘(𝑁 − 1)))
138		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 = (𝑁 − 1) → (𝑖 + 1) = ((𝑁 − 1) + 1))
139	138	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 = (𝑁 − 1) → (𝑡‘(𝑖 + 1)) = (𝑡‘((𝑁 − 1) + 1)))
140	137, 139	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 = (𝑁 − 1) → {(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} = {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))})
141	140	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = (𝑁 − 1) → ({(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ ran 𝐸))
142	141	rspcva 3280	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑁 − 1) ∈ (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ ran 𝐸)
143	136, 142	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ ∧ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ ran 𝐸)
144	41, 42	npcand 10275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁)
145	144	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℕ → (𝑡‘((𝑁 − 1) + 1)) = (𝑡‘𝑁))
146	145	preq2d 4219	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} = {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)})
147	146	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → ({(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ ran 𝐸 ↔ {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
148	147	biimpd 218	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → ({(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ ran 𝐸 → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
149	148	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ ∧ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → ({(𝑡‘(𝑁 − 1)), (𝑡‘((𝑁 − 1) + 1))} ∈ ran 𝐸 → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
150	143, 149	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ ∧ ∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸)
151	150	ex 449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
152	151	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((#‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
153	135, 152	sylbid 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝑡) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
154	153	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸)))
155	154	com3r 85	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸 → ((#‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸)))
156	155	3ad2ant3 1077	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) → ((#‘𝑡) = (𝑁 + 1) → (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸)))
157	156	imp 444	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → (𝑁 ∈ ℕ → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
158	157	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
159	158	3ad2ant3 1077	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸))
160	159	imp 444	. . . . . . . . . . . . . 14 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸)
161	160	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → {(𝑡‘(𝑁 − 1)), (𝑡‘𝑁)} ∈ ran 𝐸)
162	132, 161	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → {(𝑡‘(𝑁 − 1)), (𝑡‘0)} ∈ ran 𝐸)
163	118, 162	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸)
164	163	adantl 481	. . . . . . . . . 10 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸)
165	38, 97, 164	3jca 1235	. . . . . . . . 9 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → ((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸))
166		simpl 472	. . . . . . . . 9 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)
167	165, 166	jca 553	. . . . . . . 8 ⊢ (((#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁 ∧ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → (((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))
168	34, 167	mpancom 700	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ ((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1))) ∧ ( lastS ‘𝑡) = (𝑡‘0)) → (((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))
169	168	exp31 628	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (((𝑡 ≠ ∅ ∧ 𝑡 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑡) − 1)){(𝑡‘𝑖), (𝑡‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑡) = (𝑁 + 1)) → (( lastS ‘𝑡) = (𝑡‘0) → (((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))))
170	12, 169	sylbid 229	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (𝑡 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (( lastS ‘𝑡) = (𝑡‘0) → (((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))))
171	170	imp32 448	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ (𝑡 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → (((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁))
172		isclwwlkn 26297	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ₀) → ((𝑡 substr ⟨0, 𝑁⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ ((𝑡 substr ⟨0, 𝑁⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)))
173	6, 172	syl3an3 1353	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑡 substr ⟨0, 𝑁⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ ((𝑡 substr ⟨0, 𝑁⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)))
174		isclwwlk 26296	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((𝑡 substr ⟨0, 𝑁⟩) ∈ (𝑉 ClWWalks 𝐸) ↔ ((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸)))
175	174	3adant3 1074	. . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑡 substr ⟨0, 𝑁⟩) ∈ (𝑉 ClWWalks 𝐸) ↔ ((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸)))
176	175	anbi1d 737	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (((𝑡 substr ⟨0, 𝑁⟩) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁) ↔ (((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)))
177	173, 176	bitrd 267	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑡 substr ⟨0, 𝑁⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)))
178	177	adantr 480	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ (𝑡 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → ((𝑡 substr ⟨0, 𝑁⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (((𝑡 substr ⟨0, 𝑁⟩) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(𝑡 substr ⟨0, 𝑁⟩)) − 1)){((𝑡 substr ⟨0, 𝑁⟩)‘𝑖), ((𝑡 substr ⟨0, 𝑁⟩)‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘(𝑡 substr ⟨0, 𝑁⟩)), ((𝑡 substr ⟨0, 𝑁⟩)‘0)} ∈ ran 𝐸) ∧ (#‘(𝑡 substr ⟨0, 𝑁⟩)) = 𝑁)))
179	171, 178	mpbird 246	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ (𝑡 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑡) = (𝑡‘0))) → (𝑡 substr ⟨0, 𝑁⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
180	5, 179	sylan2b 491	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ 𝑡 ∈ 𝐷) → (𝑡 substr ⟨0, 𝑁⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
181		clwwlkbij.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
182	180, 181	fmptd 6292	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → 𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ⊆ wss 3540 ∅c0 3874 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 lastS clsw 13147 substr csubstr 13150 WWalks cwwlk 26205 WWalksN cwwlkn 26206 ClWWalks cclwwlk 26276 ClWWalksN cclwwlkn 26277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-substr 13158 df-wwlk 26207 df-wwlkn 26208 df-clwwlk 26279 df-clwwlkn 26280

This theorem is referenced by: clwwlkf1 26324 clwwlkfo 26325

W3C validator