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Theorem wwlksnextwrd 27015
Description: Lemma for wwlksnextbij 27020. (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 1081 . . . . 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 877 . . . 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 26945 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
6 simpl 474 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸))) → 𝑁 ∈ ℕ0)
7 simpl 474 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
8 nn0re 11493 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
9 2re 11282 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
109a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
11 nn0ge0 11510 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
12 2pos 11304 . . . . . . . . . . . . . . . . . . . . 21 0 < 2
1312a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < 2)
148, 10, 11, 13addgegt0d 10793 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1514adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
16 breq2 4808 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑤) = (𝑁 + 2) → (0 < (♯‘𝑤) ↔ 0 < (𝑁 + 2)))
1716ad2antll 767 . . . . . . . . . . . . . . . . . 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 13348 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word 𝑉 ∧ 0 < (♯‘𝑤)) → 𝑤 ≠ ∅)
207, 18, 19syl2an2 910 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
21 lswcl 13542 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → (lastS‘𝑤) ∈ 𝑉)
227, 20, 21syl2an2 910 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (lastS‘𝑤) ∈ 𝑉)
2322adantrr 755 . . . . . . . . . . . . . 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 13618 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ Word 𝑉 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉)
25 eleq1 2827 . . . . . . . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . . . . . . 19 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2827adantr 472 . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 6820 . . . . . . . . . . . . . . . . . 18 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩))
3433eqcoms 2768 . . . . . . . . . . . . . . . . 17 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩))
3534adantr 472 . . . . . . . . . . . . . . . 16 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) → (𝑊 ++ ⟨“(lastS‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩))
3635ad2antll 767 . . . . . . . . . . . . . . 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 6820 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑤) = (𝑁 + 2) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
3837adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2)) → ((♯‘𝑤) − 1) = ((𝑁 + 2) − 1))
39 nn0cn 11494 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
40 2cnd 11285 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
41 1cnd 10248 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4239, 40, 41addsubassd 10604 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
43 2m1e1 11327 . . . . . . . . . . . . . . . . . . . . . . . 24 (2 − 1) = 1
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (2 − 1) = 1)
4544oveq2d 6829 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
4642, 45eqtrd 2794 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
4738, 46sylan9eqr 2816 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((♯‘𝑤) − 1) = (𝑁 + 1))
4847opeq2d 4560 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ⟨0, ((♯‘𝑤) − 1)⟩ = ⟨0, (𝑁 + 1)⟩)
4948oveq2d 6829 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → (𝑤 substr ⟨0, ((♯‘𝑤) − 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
5049oveq1d 6828 . . . . . . . . . . . . . . . . 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 13668 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((♯‘𝑤) − 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
527, 20, 51syl2an2 910 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((♯‘𝑤) − 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
5350, 52eqtr3d 2796 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“(lastS‘𝑤)”⟩) = 𝑤)
5453adantrr 755 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . 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 824 . . . . . . . . . . . . . 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 27012 . . . . . . . . . . . . . 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 1484 . . . . . . . . . . . . 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 653 . . . . . . . . . . . 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 1129 . . . . . . . . . 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 26946 . . . . . . . . . . . 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 10254 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
68 1p1e2 11326 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
6968a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (1 + 1) = 2)
7069oveq2d 6829 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
7167, 70eqtrd 2794 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
7271eqeq2d 2770 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → ((♯‘𝑤) = ((𝑁 + 1) + 1) ↔ (♯‘𝑤) = (𝑁 + 2)))
7372biimpd 219 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((♯‘𝑤) = ((𝑁 + 1) + 1) → (♯‘𝑤) = (𝑁 + 2)))
7473adantr 472 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (♯‘𝑤) = (𝑁 + 2)))
77 simpl 474 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
7876, 77jctild 567 . . . . . . . . . . . . 13 ((𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = (𝑁 + 2))))
79783adant3 1127 . . . . . . . . . . . 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 1125 . . . . . . . . 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 472 . . . . . . 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 675 . . . 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 3326 . 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 2806 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 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  {crab 3054  Vcvv 3340  c0 4058  {cpr 4323  cop 4327   class class class wbr 4804  cfv 6049  (class class class)co 6813  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   < clt 10266  cmin 10458  2c2 11262  0cn0 11484  ..^cfzo 12659  chash 13311  Word cword 13477  lastSclsw 13478   ++ cconcat 13479  ⟨“cs1 13480   substr csubstr 13481  Vtxcvtx 26073  Edgcedg 26138   WWalksN cwwlksn 26929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-lsw 13486  df-concat 13487  df-s1 13488  df-substr 13489  df-wwlks 26933  df-wwlksn 26934
This theorem is referenced by:  wwlksnextsur  27018  wwlksnextbij  27020
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