Theorem wwlksnextwrd 29877

Description: Lemma for wwlksnextbij 29882. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)

Hypotheses

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

Assertion

Ref	Expression
wwlksnextwrd	⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (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 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
2		3anass 1094	. . . . 5 ⊢ (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑤) = (𝑁 + 2) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)))
3	2	bianass 642	. . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)))
4		wwlksnextbij0.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺)
5	4	wwlknbp 29822	. . . . . . . . . 10 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉))
6		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑁 ∈ ℕ0)
7		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
8		nn0re 12427	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
9		2re 12236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
10	9	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
11		nn0ge0 12443	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12		2pos 12265	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 < 2
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
14	8, 10, 11, 13	addgegt0d 11727	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
15	14	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
16		breq2 5106	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑤) = (𝑁 + 2) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
17	16	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
18	15, 17	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (♯‘𝑤))
19		hashgt0n0 14306	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 0 < (♯‘𝑤)) → 𝑤 ≠ ∅)
20	7, 18, 19	syl2an2 686	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
21		lswcl 14509	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → (lastS‘𝑤) ∈ 𝑉)
22	7, 20, 21	syl2an2 686	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (lastS‘𝑤) ∈ 𝑉)
23	22	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → (lastS‘𝑤) ∈ 𝑉)
24		pfxcl 14618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ Word 𝑉 → (𝑤 prefix (𝑁 + 1)) ∈ Word 𝑉)
25		eleq1 2816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 prefix (𝑁 + 1)) ∈ Word 𝑉))
26	24, 25	imbitrrid 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
27	26	eqcoms 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 prefix (𝑁 + 1)) = 𝑊 → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
28	27	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
29	28	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ Word 𝑉 → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
30	29	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
31	30	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → 𝑊 ∈ Word 𝑉)
32	31	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑊 ∈ Word 𝑉)
33		oveq1 7376	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
34	33	eqcoms 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 prefix (𝑁 + 1)) = 𝑊 → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
36	35	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
37		oveq1 7376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) = (𝑁 + 2) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
38	37	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
39		nn0cn 12428	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
40		2cnd 12240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
41		1cnd 11145	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
42	39, 40, 41	addsubassd 11529	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
43		2m1e1 12283	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 − 1) = 1
44	43	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → (2 − 1) = 1)
45	44	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
46	42, 45	eqtrd 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
47	38, 46	sylan9eqr 2786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((♯‘𝑤) − 1) = (𝑁 + 1))
48	47	oveq2d 7385	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (𝑤 prefix ((♯‘𝑤) − 1)) = (𝑤 prefix (𝑁 + 1)))
49	48	oveq1d 7384	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
50		pfxlswccat 14654	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
51	7, 20, 50	syl2an2 686	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
52	49, 51	eqtr3d 2766	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
53	52	adantrr 717	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
54	36, 53	eqtr2d 2765	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑤 = (𝑊 ++ ⟨“(lastS‘𝑤)”⟩))
55		simprrr 781	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)
56		wwlksnextbij0.e	. . . . . . . . . . . . . . 15 ⊢ 𝐸 = (Edg‘𝐺)
57	4, 56	wwlksnextbi 29874	. . . . . . . . . . . . . 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)))
58	6, 23, 32, 54, 55, 57	syl23anc 1379	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑊 ∈ (𝑁 WWalksN 𝐺)))
59	58	exbiri 810	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
60	59	com23 86	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
61	60	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
62	5, 61	mpcom 38	. . . . . . . . 9 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)))
63	62	expcomd 416	. . . . . . . 8 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
64	63	imp 406	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)))
65	4, 56	wwlknp 29823	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸))
66	39, 41, 41	addassd 11172	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
67		1p1e2 12282	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) = 2
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (1 + 1) = 2)
69	68	oveq2d 7385	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
70	66, 69	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
71	70	eqeq2d 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → ((♯‘𝑤) = ((𝑁 + 1) + 1) ↔ (♯‘𝑤) = (𝑁 + 2)))
72	71	biimpd 229	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → ((♯‘𝑤) = ((𝑁 + 1) + 1) → (♯‘𝑤) = (𝑁 + 2)))
73	72	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → ((♯‘𝑤) = ((𝑁 + 1) + 1) → (♯‘𝑤) = (𝑁 + 2)))
74	73	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑤) = ((𝑁 + 1) + 1) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (♯‘𝑤) = (𝑁 + 2)))
75	74	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (♯‘𝑤) = (𝑁 + 2)))
76		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
77	75, 76	jctild 525	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
78	77	3adant3 1132	. . . . . . . . . . . 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))))
79	65, 78	syl 17	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
80	79	com12 32	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
81	80	3adant1 1130	. . . . . . . . 9 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
82	5, 81	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
83	82	adantr 480	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
84	64, 83	impbid 212	. . . . . 6 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺)))
85	84	ex 412	. . . . 5 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺))))
86	85	pm5.32rd 578	. . . 4 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) ↔ (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))))
87	3, 86	bitrid 283	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) ↔ (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))))
88	87	rabbidva2 3404	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
89	1, 88	eqtrid 2776	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3402  Vcvv 3444  ∅c0 4292  {cpr 4587   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  ℝcr 11043  0cc0 11044  1c1 11045   + caddc 11047   < clt 11184   − cmin 11381  2c2 12217  ℕ₀cn0 12418  ..^cfzo 13591  ♯chash 14271  Word cword 14454  lastSclsw 14503   ++ cconcat 14511  ⟨“cs1 14536   prefix cpfx 14611  Vtxcvtx 28976  Edgcedg 29027   WWalksN cwwlksn 29806

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-xnn0 12492  df-z 12506  df-uz 12770  df-fz 13445  df-fzo 13592  df-hash 14272  df-word 14455  df-lsw 14504  df-concat 14512  df-s1 14537  df-substr 14582  df-pfx 14612  df-wwlks 29810  df-wwlksn 29811

This theorem is referenced by:  wwlksnextsurj  29880  wwlksnextbij  29882