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

Theorem wwlksnextwrd 27681
Description: Lemma for wwlksnextbij 27686. (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.
StepHypRef Expression
1 wwlksnextbij0.d . 2 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
2 3anass 1092 . . . . 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)))
32bianass 641 . . . 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)
54wwlknbp 27626 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
6 simpl 486 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑁 ∈ ℕ0)
7 simpl 486 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
8 nn0re 11894 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
9 2re 11699 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
109a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
11 nn0ge0 11910 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12 2pos 11728 . . . . . . . . . . . . . . . . . . . . 21 0 < 2
1312a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < 2)
148, 10, 11, 13addgegt0d 11202 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1514adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
16 breq2 5046 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑤) = (𝑁 + 2) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
1716ad2antll 728 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
1815, 17mpbird 260 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (♯‘𝑤))
19 hashgt0n0 13722 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word 𝑉 ∧ 0 < (♯‘𝑤)) → 𝑤 ≠ ∅)
207, 18, 19syl2an2 685 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
21 lswcl 13911 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → (lastS‘𝑤) ∈ 𝑉)
227, 20, 21syl2an2 685 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (lastS‘𝑤) ∈ 𝑉)
2322adantrr 716 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → (lastS‘𝑤) ∈ 𝑉)
24 pfxcl 14030 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ Word 𝑉 → (𝑤 prefix (𝑁 + 1)) ∈ Word 𝑉)
25 eleq1 2901 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 prefix (𝑁 + 1)) ∈ Word 𝑉))
2624, 25syl5ibr 249 . . . . . . . . . . . . . . . . . . . 20 (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2726eqcoms 2830 . . . . . . . . . . . . . . . . . . 19 ((𝑤 prefix (𝑁 + 1)) = 𝑊 → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2827adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2928com12 32 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ Word 𝑉 → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
3029adantr 484 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
3130imp 410 . . . . . . . . . . . . . . 15 (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → 𝑊 ∈ Word 𝑉)
3231adantl 485 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑊 ∈ Word 𝑉)
33 oveq1 7147 . . . . . . . . . . . . . . . . . 18 (𝑊 = (𝑤 prefix (𝑁 + 1)) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
3433eqcoms 2830 . . . . . . . . . . . . . . . . 17 ((𝑤 prefix (𝑁 + 1)) = 𝑊 → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
3534adantr 484 . . . . . . . . . . . . . . . 16 (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
3635ad2antll 728 . . . . . . . . . . . . . . 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 7147 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑤) = (𝑁 + 2) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
3837adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
39 nn0cn 11895 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
40 2cnd 11703 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
41 1cnd 10625 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4239, 40, 41addsubassd 11006 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
43 2m1e1 11751 . . . . . . . . . . . . . . . . . . . . . . 23 (2 − 1) = 1
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (2 − 1) = 1)
4544oveq2d 7156 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
4642, 45eqtrd 2857 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
4738, 46sylan9eqr 2879 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((♯‘𝑤) − 1) = (𝑁 + 1))
4847oveq2d 7156 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (𝑤 prefix ((♯‘𝑤) − 1)) = (𝑤 prefix (𝑁 + 1)))
4948oveq1d 7155 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩))
50 pfxlswccat 14066 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
517, 20, 50syl2an2 685 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix ((♯‘𝑤) − 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
5249, 51eqtr3d 2859 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 prefix (𝑁 + 1)) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
5352adantrr 716 . . . . . . . . . . . . . . 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)
5436, 53eqtr2d 2858 . . . . . . . . . . . . . 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)
574, 56wwlksnextbi 27678 . . . . . . . . . . . . . 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)))
586, 23, 32, 54, 55, 57syl23anc 1374 . . . . . . . . . . . . 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)))
5958exbiri 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))))
6059com23 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))))
61603ad2ant2 1131 . . . . . . . . . 10 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ 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))))
625, 61mpcom 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)))
6362expcomd 420 . . . . . . . 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))))
6463imp 410 . . . . . . 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)))
654, 56wwlknp 27627 . . . . . . . . . . . 12 (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸))
6639, 41, 41addassd 10652 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
67 1p1e2 11750 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
6867a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (1 + 1) = 2)
6968oveq2d 7156 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
7066, 69eqtrd 2857 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
7170eqeq2d 2833 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → ((♯‘𝑤) = ((𝑁 + 1) + 1) ↔ (♯‘𝑤) = (𝑁 + 2)))
7271biimpd 232 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((♯‘𝑤) = ((𝑁 + 1) + 1) → (♯‘𝑤) = (𝑁 + 2)))
7372adantr 484 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → ((♯‘𝑤) = ((𝑁 + 1) + 1) → (♯‘𝑤) = (𝑁 + 2)))
7473com12 32 . . . . . . . . . . . . . . 15 ((♯‘𝑤) = ((𝑁 + 1) + 1) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (♯‘𝑤) = (𝑁 + 2)))
7574adantl 485 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (♯‘𝑤) = (𝑁 + 2)))
76 simpl 486 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
7775, 76jctild 529 . . . . . . . . . . . . 13 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
78773adant3 1129 . . . . . . . . . . . 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))))
7965, 78syl 17 . . . . . . . . . . 11 (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
8079com12 32 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
81803adant1 1127 . . . . . . . . 9 ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
825, 81syl 17 . . . . . . . 8 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
8382adantr 484 . . . . . . 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))))
8464, 83impbid 215 . . . . . 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)))
8584ex 416 . . . . 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))))
8685pm5.32rd 581 . . . 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))))
873, 86syl5bb 286 . . 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))))
8887rabbidva2 3451 . 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)})
891, 88syl5eq 2869 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011  wral 3130  {crab 3134  Vcvv 3469  c0 4265  {cpr 4541   class class class wbr 5042  cfv 6334  (class class class)co 7140  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664  cmin 10859  2c2 11680  0cn0 11885  ..^cfzo 13028  chash 13686  Word cword 13857  lastSclsw 13905   ++ cconcat 13913  ⟨“cs1 13940   prefix cpfx 14023  Vtxcvtx 26787  Edgcedg 26838   WWalksN cwwlksn 27610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-lsw 13906  df-concat 13914  df-s1 13941  df-substr 13994  df-pfx 14024  df-wwlks 27614  df-wwlksn 27615
This theorem is referenced by:  wwlksnextsurj  27684  wwlksnextbij  27686
  Copyright terms: Public domain W3C validator