MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlksnextwrd Structured version   Visualization version   GIF version

Theorem wwlksnextwrd 26661
Description: Lemma for wwlksnextbij 26666. (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.
StepHypRef 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 1040 . . . . 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)))
32bianass 841 . . . 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)
54wwlknbp 26602 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ 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 11245 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
9 2re 11034 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
109a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
11 nn0ge0 11262 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12 2pos 11056 . . . . . . . . . . . . . . . . . . . . 21 0 < 2
1312a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < 2)
148, 10, 11, 13addgegt0d 10545 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1514adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
16 breq2 4617 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑤) = (𝑁 + 2) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
1716ad2antll 764 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
1815, 17mpbird 247 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (#‘𝑤))
19 hashgt0n0 13096 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word 𝑉 ∧ 0 < (#‘𝑤)) → 𝑤 ≠ ∅)
207, 18, 19syl2an2 874 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
21 lswcl 13294 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ( lastS ‘𝑤) ∈ 𝑉)
227, 20, 21syl2an2 874 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ( lastS ‘𝑤) ∈ 𝑉)
2322adantrr 752 . . . . . . . . . . . . . 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 13357 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ Word 𝑉 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉)
25 eleq1 2686 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉))
2624, 25syl5ibr 236 . . . . . . . . . . . . . . . . . . . 20 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2726eqcoms 2629 . . . . . . . . . . . . . . . . . . 19 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2827adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2928com12 32 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ Word 𝑉 → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
3029adantr 481 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
3130imp 445 . . . . . . . . . . . . . . 15 (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑊 ∈ Word 𝑉)
3231adantl 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 6611 . . . . . . . . . . . . . . . . . 18 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3433eqcoms 2629 . . . . . . . . . . . . . . . . 17 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3534adantr 481 . . . . . . . . . . . . . . . 16 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3635ad2antll 764 . . . . . . . . . . . . . . 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 6611 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑤) = (𝑁 + 2) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
3837adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
39 nn0cn 11246 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
40 2cnd 11037 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
41 1cnd 10000 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4239, 40, 41addsubassd 10356 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
43 2m1e1 11079 . . . . . . . . . . . . . . . . . . . . . . . 24 (2 − 1) = 1
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (2 − 1) = 1)
4544oveq2d 6620 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
4642, 45eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
4738, 46sylan9eqr 2677 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((#‘𝑤) − 1) = (𝑁 + 1))
4847opeq2d 4377 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ⟨0, ((#‘𝑤) − 1)⟩ = ⟨0, (𝑁 + 1)⟩)
4948oveq2d 6620 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
5049oveq1d 6619 . . . . . . . . . . . . . . . . 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 13406 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
527, 20, 51syl2an2 874 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
5350, 52eqtr3d 2657 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
5453adantrr 752 . . . . . . . . . . . . . . 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)
5536, 54eqtr2d 2656 . . . . . . . . . . . . . 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 804 . . . . . . . . . . . . . 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)
584, 57wwlksnextbi 26658 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ ( lastS ‘𝑤) ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑊 ∈ (𝑁 WWalksN 𝐺)))
596, 23, 32, 55, 56, 58syl23anc 1330 . . . . . . . . . . . . 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)))
6059exbiri 651 . . . . . . . . . . . 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))))
6160com23 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))))
62613ad2ant2 1081 . . . . . . . . . 10 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ 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))))
635, 62mpcom 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)))
6463expcomd 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))))
6564imp 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)))
664, 57wwlknp 26603 . . . . . . . . . . . 12 (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸))
6739, 41, 41addassd 10006 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
68 1p1e2 11078 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
6968a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (1 + 1) = 2)
7069oveq2d 6620 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
7167, 70eqtrd 2655 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
7271eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) ↔ (#‘𝑤) = (𝑁 + 2)))
7372biimpd 219 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
7473adantr 481 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
7574com12 32 . . . . . . . . . . . . . . 15 ((#‘𝑤) = ((𝑁 + 1) + 1) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
7675adantl 482 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
77 simpl 473 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
7876, 77jctild 565 . . . . . . . . . . . . 13 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
79783adant3 1079 . . . . . . . . . . . 12 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
8066, 79syl 17 . . . . . . . . . . 11 (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
8180com12 32 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
82813adant1 1077 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
835, 82syl 17 . . . . . . . 8 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
8483adantr 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))))
8565, 84impbid 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)))
8685ex 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))))
8786pm5.32rd 671 . . . 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))))
883, 87syl5bb 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))))
8988rabbidva2 3174 . 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)})
901, 89syl5eq 2667 1 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  {crab 2911  Vcvv 3186  c0 3891  {cpr 4150  cop 4154   class class class wbr 4613  cfv 5847  (class class class)co 6604  cr 9879  0cc0 9880  1c1 9881   + caddc 9883   < clt 10018  cmin 10210  2c2 11014  0cn0 11236  ..^cfzo 12406  #chash 13057  Word cword 13230   lastS clsw 13231   ++ cconcat 13232  ⟨“cs1 13233   substr csubstr 13234  Vtxcvtx 25774  Edgcedg 25839   WWalksN cwwlksn 26587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-wwlks 26591  df-wwlksn 26592
This theorem is referenced by:  wwlksnextsur  26664  wwlksnextbij  26666
  Copyright terms: Public domain W3C validator