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Theorem wwlksnextwrdOLD 27174
Description: Obsolete version of wwlksnextwrd 27169 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 18-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
wwlksnextbij0OLD.v 𝑉 = (Vtx‘𝐺)
wwlksnextbij0OLD.e 𝐸 = (Edg‘𝐺)
wwlksnextbij0OLD.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
Assertion
Ref Expression
wwlksnextwrdOLD (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 wwlksnextwrdOLD
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlksnextbij0OLD.d . 2 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
2 3anass 1117 . . . . 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 633 . . . 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 wwlksnextbij0OLD.v . . . . . . . . . . 11 𝑉 = (Vtx‘𝐺)
54wwlknbp 27093 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
6 simpl 475 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑁 ∈ ℕ0)
7 simpl 475 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
8 nn0re 11590 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
9 2re 11387 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
109a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
11 nn0ge0 11607 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12 2pos 11423 . . . . . . . . . . . . . . . . . . . . 21 0 < 2
1312a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < 2)
148, 10, 11, 13addgegt0d 10893 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1514adantr 473 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
16 breq2 4847 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑤) = (𝑁 + 2) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
1716ad2antll 721 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
1815, 17mpbird 249 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (♯‘𝑤))
19 hashgt0n0 13406 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word 𝑉 ∧ 0 < (♯‘𝑤)) → 𝑤 ≠ ∅)
207, 18, 19syl2an2 678 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
21 lswcl 13588 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → (lastS‘𝑤) ∈ 𝑉)
227, 20, 21syl2an2 678 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (lastS‘𝑤) ∈ 𝑉)
2322adantrr 709 . . . . . . . . . . . . . 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 13669 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ Word 𝑉 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉)
25 eleq1 2866 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉))
2624, 25syl5ibr 238 . . . . . . . . . . . . . . . . . . . 20 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2726eqcoms 2807 . . . . . . . . . . . . . . . . . . 19 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2827adantr 473 . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → 𝑊 ∈ Word 𝑉))
3130imp 396 . . . . . . . . . . . . . . 15 (((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)) → 𝑊 ∈ Word 𝑉)
3231adantl 474 . . . . . . . . . . . . . 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 6885 . . . . . . . . . . . . . . . . . 18 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩))
3433eqcoms 2807 . . . . . . . . . . . . . . . . 17 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩))
3534adantr 473 . . . . . . . . . . . . . . . 16 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩))
3635ad2antll 721 . . . . . . . . . . . . . . 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 6885 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑤) = (𝑁 + 2) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
3837adantl 474 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
39 nn0cn 11591 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
40 2cnd 11391 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
41 1cnd 10323 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4239, 40, 41addsubassd 10704 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
43 2m1e1 11446 . . . . . . . . . . . . . . . . . . . . . . . 24 (2 − 1) = 1
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (2 − 1) = 1)
4544oveq2d 6894 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
4642, 45eqtrd 2833 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
4738, 46sylan9eqr 2855 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((♯‘𝑤) − 1) = (𝑁 + 1))
4847opeq2d 4600 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ⟨0, ((♯‘𝑤) − 1)⟩ = ⟨0, (𝑁 + 1)⟩)
4948oveq2d 6894 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (𝑤 substr ⟨0, ((♯‘𝑤) − 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
5049oveq1d 6893 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((♯‘𝑤) − 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩))
51 swrdccatwrdOLD 13764 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((♯‘𝑤) − 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
527, 20, 51syl2an2 678 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((♯‘𝑤) − 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
5350, 52eqtr3d 2835 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
5453adantrr 709 . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . 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 801 . . . . . . . . . . . . . 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 wwlksnextbij0OLD.e . . . . . . . . . . . . . . 15 𝐸 = (Edg‘𝐺)
584, 57wwlksnextbi 27163 . . . . . . . . . . . . . 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 1497 . . . . . . . . . . . . 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 846 . . . . . . . . . . . 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 1165 . . . . . . . . . 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 407 . . . . . . . 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 396 . . . . . . 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 27094 . . . . . . . . . . . 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 10351 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
68 1p1e2 11445 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
6968a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (1 + 1) = 2)
7069oveq2d 6894 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
7167, 70eqtrd 2833 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
7271eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → ((♯‘𝑤) = ((𝑁 + 1) + 1) ↔ (♯‘𝑤) = (𝑁 + 2)))
7372biimpd 221 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((♯‘𝑤) = ((𝑁 + 1) + 1) → (♯‘𝑤) = (𝑁 + 2)))
7473adantr 473 . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (♯‘𝑤) = (𝑁 + 2)))
77 simpl 475 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
7876, 77jctild 522 . . . . . . . . . . . . 13 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
79783adant3 1163 . . . . . . . . . . . 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 1161 . . . . . . . . 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 473 . . . . . . 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 204 . . . . . 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 402 . . . . 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 574 . . . 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 275 . . 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 3370 . 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 2845 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 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  {crab 3093  Vcvv 3385  c0 4115  {cpr 4370  cop 4374   class class class wbr 4843  cfv 6101  (class class class)co 6878  cr 10223  0cc0 10224  1c1 10225   + caddc 10227   < clt 10363  cmin 10556  2c2 11368  0cn0 11580  ..^cfzo 12720  chash 13370  Word cword 13534  lastSclsw 13582   ++ cconcat 13590  ⟨“cs1 13615   substr csubstr 13664  Vtxcvtx 26231  Edgcedg 26282   WWalksN cwwlksn 27077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-n0 11581  df-xnn0 11653  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-lsw 13583  df-concat 13591  df-s1 13616  df-substr 13665  df-pfx 13714  df-wwlks 27081  df-wwlksn 27082
This theorem is referenced by:  wwlksnextsurOLD  27177  wwlksnextbijOLD  27180
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