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Theorem wwlksnextwrd 29418

Description: Lemma for wwlksnextbij 29423. (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 1093	. . . . 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 638	. . . 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 29363	. . . . . . . . . 10 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉))
6		simpl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑁 ∈ ℕ₀)
7		simpl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
8		nn0re 12485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
9		2re 12290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
10	9	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ)
11		nn0ge0 12501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
12		2pos 12319	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 < 2
13	12	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 0 < 2)
14	8, 10, 11, 13	addgegt0d 11791	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 0 < (𝑁 + 2))
15	14	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
16		breq2 5151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑤) = (𝑁 + 2) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
17	16	ad2antll 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
18	15, 17	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (♯‘𝑤))
19		hashgt0n0 14329	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 0 < (♯‘𝑤)) → 𝑤 ≠ ∅)
20	7, 18, 19	syl2an2 682	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
21		lswcl 14522	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → (lastS‘𝑤) ∈ 𝑉)
22	7, 20, 21	syl2an2 682	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (lastS‘𝑤) ∈ 𝑉)
23	22	adantrr 713	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → (lastS‘𝑤) ∈ 𝑉)
24		pfxcl 14631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ Word 𝑉 → (𝑤 prefix (𝑁 + 1)) ∈ Word 𝑉)
25		eleq1 2819	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 prefix (𝑁 + 1)) ∈ Word 𝑉))
26	24, 25	imbitrrid 245	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
27	26	eqcoms 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 prefix (𝑁 + 1)) = 𝑊 → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
28	27	adantr 479	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
31	30	imp 405	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → 𝑊 ∈ Word 𝑉)
32	31	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑊 ∈ Word 𝑉)
33		oveq1 7418	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
34	33	eqcoms 2738	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 prefix (𝑁 + 1)) = 𝑊 → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
35	34	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
36	35	ad2antll 725	. . . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) = (𝑁 + 2) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
38	37	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
39		nn0cn 12486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
40		2cnd 12294	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℂ)
41		1cnd 11213	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
42	39, 40, 41	addsubassd 11595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
43		2m1e1 12342	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 − 1) = 1
44	43	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → (2 − 1) = 1)
45	44	oveq2d 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + (2 − 1)) = (𝑁 + 1))
46	42, 45	eqtrd 2770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 2) − 1) = (𝑁 + 1))
47	38, 46	sylan9eqr 2792	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((♯‘𝑤) − 1) = (𝑁 + 1))
48	47	oveq2d 7427	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (𝑤 prefix ((♯‘𝑤) − 1)) = (𝑤 prefix (𝑁 + 1)))
49	48	oveq1d 7426	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
50		pfxlswccat 14667	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
51	7, 20, 50	syl2an2 682	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
52	49, 51	eqtr3d 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
53	52	adantrr 713	. . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . 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 29415	. . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . 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 807	. . . . . . . . . . . 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 1132	. . . . . . . . . 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 415	. . . . . . . 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 405	. . . . . . 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 29364	. . . . . . . . . . . 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 11240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
67		1p1e2 12341	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) = 2
68	67	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (1 + 1) = 2)
69	68	oveq2d 7427	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + (1 + 1)) = (𝑁 + 2))
70	66, 69	eqtrd 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) = (𝑁 + 2))
71	70	eqeq2d 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → ((♯‘𝑤) = ((𝑁 + 1) + 1) ↔ (♯‘𝑤) = (𝑁 + 2)))
72	71	biimpd 228	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → ((♯‘𝑤) = ((𝑁 + 1) + 1) → (♯‘𝑤) = (𝑁 + 2)))
73	72	adantr 479	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉) → (♯‘𝑤) = (𝑁 + 2)))
76		simpl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
77	75, 76	jctild 524	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
78	77	3adant3 1130	. . . . . . . . . . . 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 1128	. . . . . . . . 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 479	. . . . . . 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 211	. . . . . 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 411	. . . . 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 576	. . . 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 282	. . 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 3432	. 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 2782	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 {crab 3430 Vcvv 3472 ∅c0 4321 {cpr 4629 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 − cmin 11448 2c2 12271 ℕ₀cn0 12476 ..^cfzo 13631 ♯chash 14294 Word cword 14468 lastSclsw 14516 ++ cconcat 14524 ⟨“cs1 14549 prefix cpfx 14624 Vtxcvtx 28523 Edgcedg 28574 WWalksN cwwlksn 29347

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-lsw 14517 df-concat 14525 df-s1 14550 df-substr 14595 df-pfx 14625 df-wwlks 29351 df-wwlksn 29352

This theorem is referenced by: wwlksnextsurj 29421 wwlksnextbij 29423

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