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Theorem wwlknextbi 26253
 Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
Assertion
Ref Expression
wwlknextbi (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Proof of Theorem wwlknextbi
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlknimp 26215 . . . 4 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
2 wwlknred 26251 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
32ad2antrr 758 . . . . . . 7 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
4 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (#‘𝑊) = (#‘(𝑇 ++ ⟨“𝑆”⟩)))
54eqeq1d 2612 . . . . . . . . . . . . . . . 16 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
653ad2ant2 1076 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
76adantl 481 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
8 s1cl 13235 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
98adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0𝑆𝑉) → ⟨“𝑆”⟩ ∈ Word 𝑉)
109anim2i 591 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0𝑆𝑉)) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
1110ancoms 468 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
12 ccatlen 13213 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
1311, 12syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
1413eqeq1d 2612 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
15 s1len 13238 . . . . . . . . . . . . . . . . . . . 20 (#‘⟨“𝑆”⟩) = 1
1615a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (#‘⟨“𝑆”⟩) = 1)
1716oveq2d 6565 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((#‘𝑇) + 1))
1817eqeq1d 2612 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((#‘𝑇) + 1) = ((𝑁 + 1) + 1)))
19 lencl 13179 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ Word 𝑉 → (#‘𝑇) ∈ ℕ0)
2019nn0cnd 11230 . . . . . . . . . . . . . . . . . . 19 (𝑇 ∈ Word 𝑉 → (#‘𝑇) ∈ ℂ)
2120adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (#‘𝑇) ∈ ℂ)
22 peano2nn0 11210 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
2322nn0cnd 11230 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
2423ad2antrr 758 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ)
25 1cnd 9935 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ)
2621, 24, 25addcan2d 10119 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((#‘𝑇) + 1) = ((𝑁 + 1) + 1) ↔ (#‘𝑇) = (𝑁 + 1)))
2714, 18, 263bitrd 293 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ (#‘𝑇) = (𝑁 + 1)))
28 opeq2 4341 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 + 1) = (#‘𝑇) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑇)⟩)
2928eqcoms 2618 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑇) = (𝑁 + 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑇)⟩)
3029oveq2d 6565 . . . . . . . . . . . . . . . . . 18 ((#‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩))
31 swrdccat1 13309 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩) = 𝑇)
3211, 31syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩) = 𝑇)
3330, 32sylan9eqr 2666 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (#‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
3433ex 449 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
3527, 34sylbid 229 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
36353ad2antr1 1219 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
377, 36sylbid 229 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
3837imp 444 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
39 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩))
4039eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
41403ad2ant2 1076 . . . . . . . . . . . . 13 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
4241ad2antlr 759 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
4338, 42mpbird 246 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
4443eleq1d 2672 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
4544biimpd 218 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
4645ex 449 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
4746com23 84 . . . . . . 7 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((#‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
483, 47syld 46 . . . . . 6 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
4948com13 86 . . . . 5 ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
50493ad2ant2 1076 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
511, 50mpcom 37 . . 3 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
5251com12 32 . 2 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
53 wwlknext 26252 . . . . . . . . . . 11 ((𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))
54 eleq1 2676 . . . . . . . . . . 11 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
5553, 54syl5ibrcom 236 . . . . . . . . . 10 ((𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
56553exp 1256 . . . . . . . . 9 (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑆𝑉 → ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))))
5756com23 84 . . . . . . . 8 (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 → (𝑆𝑉 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))))
5857com14 94 . . . . . . 7 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ({( lastS ‘𝑇), 𝑆} ∈ ran 𝐸 → (𝑆𝑉 → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))))
5958imp 444 . . . . . 6 ((𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑆𝑉 → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
60593adant1 1072 . . . . 5 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑆𝑉 → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
6160com12 32 . . . 4 (𝑆𝑉 → ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
6261adantl 481 . . 3 ((𝑁 ∈ ℕ0𝑆𝑉) → ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
6362imp 444 . 2 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
6452, 63impbid 201 1 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {cpr 4127  ⟨cop 4131  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818  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-n0 11170  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:  wwlkextwrd  26256
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