Theorem wwlksnextwrd 26792

Description: Lemma for wwlksnextbij 26797. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 18-Apr-2021.)

Hypotheses

Ref	Expression
wwlksnextbij0.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e	⊢ 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}

Assertion

Ref	Expression
wwlksnextwrd	⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑊

Allowed substitution hints:   𝐷(𝑤)   𝐸(𝑤)   𝑉(𝑤)

Proof of Theorem wwlksnextwrd

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnextbij0.d	. 2 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
2		3anass 1042	. . . . 5 ⊢ (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) ↔ ((#‘𝑤) = (𝑁 + 2) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)))
3	2	bianass 842	. . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)))
4		wwlksnextbij0.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
5	4	wwlknbp 26733	. . . . . . . . . 10 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉))
6		simpl 473	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → 𝑁 ∈ ℕ0)
7		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
8		nn0re 11301	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
9		2re 11090	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
10	9	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
11		nn0ge0 11318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12		2pos 11112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 < 2
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
14	8, 10, 11, 13	addgegt0d 10601	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
15	14	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
16		breq2 4657	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑤) = (𝑁 + 2) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
17	16	ad2antll 765	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
18	15, 17	mpbird 247	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (#‘𝑤))
19		hashgt0n0 13156	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 0 < (#‘𝑤)) → 𝑤 ≠ ∅)
20	7, 18, 19	syl2an2 875	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
21		lswcl 13355	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → ( lastS ‘𝑤) ∈ 𝑉)
22	7, 20, 21	syl2an2 875	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ( lastS ‘𝑤) ∈ 𝑉)
23	22	adantrr 753	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → ( lastS ‘𝑤) ∈ 𝑉)
24		swrdcl 13419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ Word 𝑉 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉)
25		eleq1 2689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉))
26	24, 25	syl5ibr 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
27	26	eqcoms 2630	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
28	27	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
29	28	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ Word 𝑉 → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
30	29	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
31	30	imp 445	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑊 ∈ Word 𝑉)
32	31	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → 𝑊 ∈ Word 𝑉)
33		oveq1 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
34	33	eqcoms 2630	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
35	34	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
36	35	ad2antll 765	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
37		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑤) = (𝑁 + 2) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
38	37	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
39		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
40		2cnd 11093	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
41		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
42	39, 40, 41	addsubassd 10412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
43		2m1e1 11135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 − 1) = 1
44	43	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → (2 − 1) = 1)
45	44	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
46	42, 45	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
47	38, 46	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((#‘𝑤) − 1) = (𝑁 + 1))
48	47	opeq2d 4409	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ⟨0, ((#‘𝑤) − 1)⟩ = ⟨0, (𝑁 + 1)⟩)
49	48	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
50	49	oveq1d 6665	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
51		swrdccatwrd 13468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
52	7, 20, 51	syl2an2 875	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
53	50, 52	eqtr3d 2658	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
54	53	adantrr 753	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
55	36, 54	eqtr2d 2657	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩))
56		simprrr 805	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)
57		wwlksnextbij0.e	. . . . . . . . . . . . . . 15 ⊢ 𝐸 = (Edg‘𝐺)
58	4, 57	wwlksnextbi 26789	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ ( lastS ‘𝑤) ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑊 ∈ (𝑁 WWalksN 𝐺)))
59	6, 23, 32, 55, 56, 58	syl23anc 1333	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑊 ∈ (𝑁 WWalksN 𝐺)))
60	59	exbiri 652	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
61	60	com23 86	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
62	61	3ad2ant2 1083	. . . . . . . . . 10 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
63	5, 62	mpcom 38	. . . . . . . . 9 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)))
64	63	expcomd 454	. . . . . . . 8 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
65	64	imp 445	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)))
66	4, 57	wwlknp 26734	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸))
67	39, 41, 41	addassd 10062	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
68		1p1e2 11134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) = 2
69	68	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (1 + 1) = 2)
70	69	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
71	67, 70	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
72	71	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) ↔ (#‘𝑤) = (𝑁 + 2)))
73	72	biimpd 219	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
74	73	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
75	74	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑤) = ((𝑁 + 1) + 1) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
76	75	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
77		simpl 473	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
78	76, 77	jctild 566	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
79	78	3adant3 1081	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
80	66, 79	syl 17	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
81	80	com12 32	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
82	81	3adant1 1079	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
83	5, 82	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
84	83	adantr 481	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
85	65, 84	impbid 202	. . . . . 6 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)))
86	85	ex 450	. . . . 5 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
87	86	pm5.32rd 672	. . . 4 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) ↔ (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))))
88	3, 87	syl5bb 272	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) ↔ (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))))
89	88	rabbidva2 3186	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
90	1, 89	syl5eq 2668	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916  Vcvv 3200  ∅c0 3915  {cpr 4179  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   − cmin 10266  2c2 11070  ℕ₀cn0 11292  ..^cfzo 12465  #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293  ⟨“cs1 13294   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wwlksnextsur  26795  wwlksnextbij  26797