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Theorem wwlkextwrd 26256
 Description: Lemma 0 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
Hypothesis
Ref Expression
wwlkextbij.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
Assertion
Ref Expression
wwlkextwrd (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑉   𝑤,𝑊
Allowed substitution hint:   𝐷(𝑤)

Proof of Theorem wwlkextwrd
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlkextbij.d . 2 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
2 3anass 1035 . . . . . 6 (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ((#‘𝑤) = (𝑁 + 2) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)))
32anbi2i 726 . . . . 5 ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))))
4 anass 679 . . . . 5 (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))))
53, 4bitr4i 266 . . . 4 ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)))
6 wwlknprop 26214 . . . . . . . . . 10 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉)))
7 simpl 472 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → 𝑁 ∈ ℕ0)
8 simpl 472 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
98adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 𝑤 ∈ Word 𝑉)
10 nn0re 11178 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
11 2re 10967 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
1211a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
13 nn0ge0 11195 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
14 2pos 10989 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
1514a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 < 2)
1610, 12, 13, 15addgegt0d 10480 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1716adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
18 breq2 4587 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑤) = (𝑁 + 2) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
1918ad2antll 761 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
2017, 19mpbird 246 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (#‘𝑤))
21 hashgt0n0 13017 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word 𝑉 ∧ 0 < (#‘𝑤)) → 𝑤 ≠ ∅)
229, 20, 21syl2anc 691 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
23 lswcl 13208 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ( lastS ‘𝑤) ∈ 𝑉)
249, 22, 23syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ( lastS ‘𝑤) ∈ 𝑉)
2524adantrr 749 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → ( lastS ‘𝑤) ∈ 𝑉)
26 swrdcl 13271 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 ∈ Word 𝑉 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉)
27 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉))
2826, 27syl5ibr 235 . . . . . . . . . . . . . . . . . . . . 21 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
2928eqcoms 2618 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
3029adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → (𝑤 ∈ Word 𝑉𝑊 ∈ Word 𝑉))
3130com12 32 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ Word 𝑉 → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → 𝑊 ∈ Word 𝑉))
3231adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → 𝑊 ∈ Word 𝑉))
3332imp 444 . . . . . . . . . . . . . . . 16 (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑊 ∈ Word 𝑉)
3433adantl 481 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → 𝑊 ∈ Word 𝑉)
35 oveq1 6556 . . . . . . . . . . . . . . . . . . 19 (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3635eqcoms 2618 . . . . . . . . . . . . . . . . . 18 ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3736adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
3837ad2antll 761 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
39 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑤) = (𝑁 + 2) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
4039adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
41 nn0cn 11179 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
42 2cnd 10970 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
43 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4441, 42, 43addsubassd 10291 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
45 2m1e1 11012 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2 − 1) = 1
4645a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (2 − 1) = 1)
4746oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
4844, 47eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
4940, 48sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((#‘𝑤) − 1) = (𝑁 + 1))
5049opeq2d 4347 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ⟨0, ((#‘𝑤) − 1)⟩ = ⟨0, (𝑁 + 1)⟩)
5150oveq2d 6565 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
5251oveq1d 6564 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
53 swrdccatwrd 13320 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word 𝑉𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
549, 22, 53syl2anc 691 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
5552, 54eqtr3d 2646 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
5655adantrr 749 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
5738, 56eqtr2d 2645 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩))
58 simprrr 801 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)
59 wwlknextbi 26253 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ ( lastS ‘𝑤) ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
607, 25, 34, 57, 58, 59syl23anc 1325 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
6160exbiri 650 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0 → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
6261com23 84 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
6362adantr 480 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
6463adantl 481 . . . . . . . . . 10 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
656, 64mpcom 37 . . . . . . . . 9 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
6665expcomd 453 . . . . . . . 8 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
6766imp 444 . . . . . . 7 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
68 wwlknimp 26215 . . . . . . . . . . . 12 (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))
6941, 43, 43addassd 9941 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
70 1p1e2 11011 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) = 2
7170a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (1 + 1) = 2)
7271oveq2d 6565 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
7369, 72eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
7473eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) ↔ (#‘𝑤) = (𝑁 + 2)))
7574biimpd 218 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
7675adantr 480 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
7776com12 32 . . . . . . . . . . . . . . 15 ((#‘𝑤) = ((𝑁 + 1) + 1) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
7877adantl 481 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
79 simpl 472 . . . . . . . . . . . . . 14 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
8078, 79jctild 564 . . . . . . . . . . . . 13 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
81803adant3 1074 . . . . . . . . . . . 12 ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
8268, 81syl 17 . . . . . . . . . . 11 (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
8382com12 32 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
8483adantl 481 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉)) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
856, 84syl 17 . . . . . . . 8 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
8685adantr 480 . . . . . . 7 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
8767, 86impbid 201 . . . . . 6 ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
8887ex 449 . . . . 5 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
8988pm5.32rd 670 . . . 4 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))))
905, 89syl5bb 271 . . 3 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))))
9190rabbidva2 3162 . 2 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})
921, 91syl5eq 2656 1 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  2c2 10947  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150   WWalksN cwwlkn 26206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wwlkextsur  26259  wwlkextbij  26261
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