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Theorem wwlksnextwrd 26860
Description: Lemma for wwlksnextbij 26865. (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 1059 . . . . 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 859 . . . 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 26790 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
6 simpl 472 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))) → 𝑁 ∈ ℕ0)
7 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
8 nn0re 11339 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
9 2re 11128 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
109a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
11 nn0ge0 11356 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12 2pos 11150 . . . . . . . . . . . . . . . . . . . . 21 0 < 2
1312a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < 2)
148, 10, 11, 13addgegt0d 10639 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1514adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
16 breq2 4689 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑤) = (𝑁 + 2) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
1716ad2antll 765 . . . . . . . . . . . . . . . . . 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 13194 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word 𝑉 ∧ 0 < (#‘𝑤)) → 𝑤 ≠ ∅)
207, 18, 19syl2an2 892 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
21 lswcl 13388 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ( lastS ‘𝑤) ∈ 𝑉)
227, 20, 21syl2an2 892 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ( lastS ‘𝑤) ∈ 𝑉)
2322adantrr 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 13464 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ Word 𝑉 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉)
25 eleq1 2718 . . . . . . . . . . . . . . . . . . . . 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 2659 . . . . . . . . . . . . . . . . . . 19 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2827adantr 480 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
3130imp 444 . . . . . . . . . . . . . . 15 (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)) → 𝑊 ∈ Word 𝑉)
3231adantl 481 . . . . . . . . . . . . . 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 6697 . . . . . . . . . . . . . . . . . 18 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3433eqcoms 2659 . . . . . . . . . . . . . . . . 17 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3534adantr 480 . . . . . . . . . . . . . . . 16 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3635ad2antll 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 6697 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑤) = (𝑁 + 2) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
3837adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
39 nn0cn 11340 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
40 2cnd 11131 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
41 1cnd 10094 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4239, 40, 41addsubassd 10450 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
43 2m1e1 11173 . . . . . . . . . . . . . . . . . . . . . . . 24 (2 − 1) = 1
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (2 − 1) = 1)
4544oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
4642, 45eqtrd 2685 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
4738, 46sylan9eqr 2707 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((#‘𝑤) − 1) = (𝑁 + 1))
4847opeq2d 4440 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ⟨0, ((#‘𝑤) − 1)⟩ = ⟨0, (𝑁 + 1)⟩)
4948oveq2d 6706 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
5049oveq1d 6705 . . . . . . . . . . . . . . . . 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 13514 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
527, 20, 51syl2an2 892 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
5350, 52eqtr3d 2687 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
5453adantrr 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)
5536, 54eqtr2d 2686 . . . . . . . . . . . . . 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 822 . . . . . . . . . . . . . 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 26857 . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 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 1103 . . . . . . . . . 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 453 . . . . . . . 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 444 . . . . . . 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 26791 . . . . . . . . . . . 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 10100 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
68 1p1e2 11172 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
6968a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (1 + 1) = 2)
7069oveq2d 6706 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
7167, 70eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
7271eqeq2d 2661 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) ↔ (#‘𝑤) = (𝑁 + 2)))
7372biimpd 219 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
7473adantr 480 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
77 simpl 472 . . . . . . . . . . . . . 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 1101 . . . . . . . . . . . 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 1099 . . . . . . . . 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 480 . . . . . . 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 449 . . . . 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 673 . . . 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 3217 . 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 2697 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  Vcvv 3231  c0 3948  {cpr 4212  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cmin 10304  2c2 11108  0cn0 11330  ..^cfzo 12504  #chash 13157  Word cword 13323   lastS clsw 13324   ++ cconcat 13325  ⟨“cs1 13326   substr csubstr 13327  Vtxcvtx 25919  Edgcedg 25984   WWalksN cwwlksn 26774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-wwlks 26778  df-wwlksn 26779
This theorem is referenced by:  wwlksnextsur  26863  wwlksnextbij  26865
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