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Theorem wwlksnextbi 26658
Description: Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.)
Hypotheses
Ref Expression
wwlksnext.v 𝑉 = (Vtx‘𝐺)
wwlksnext.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
wwlksnextbi (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))

Proof of Theorem wwlksnextbi
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlksnext.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 wwlksnext.e . . . . 5 𝐸 = (Edg‘𝐺)
31, 2wwlknp 26603 . . . 4 (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
4 wwlksnred 26656 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalksN 𝐺)))
54ad2antrr 761 . . . . . . 7 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalksN 𝐺)))
6 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (#‘𝑊) = (#‘(𝑇 ++ ⟨“𝑆”⟩)))
76eqeq1d 2623 . . . . . . . . . . . . . . . 16 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
873ad2ant2 1081 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
98adantl 482 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
10 s1cl 13321 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
1110adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0𝑆𝑉) → ⟨“𝑆”⟩ ∈ Word 𝑉)
1211anim2i 592 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0𝑆𝑉)) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
1312ancoms 469 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
14 ccatlen 13299 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
1513, 14syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((#‘𝑇) + (#‘⟨“𝑆”⟩)))
1615eqeq1d 2623 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
17 s1len 13324 . . . . . . . . . . . . . . . . . . . 20 (#‘⟨“𝑆”⟩) = 1
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (#‘⟨“𝑆”⟩) = 1)
1918oveq2d 6620 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((#‘𝑇) + 1))
2019eqeq1d 2623 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((#‘𝑇) + (#‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((#‘𝑇) + 1) = ((𝑁 + 1) + 1)))
21 lencl 13263 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ Word 𝑉 → (#‘𝑇) ∈ ℕ0)
2221nn0cnd 11297 . . . . . . . . . . . . . . . . . . 19 (𝑇 ∈ Word 𝑉 → (#‘𝑇) ∈ ℂ)
2322adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (#‘𝑇) ∈ ℂ)
24 peano2nn0 11277 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
2524nn0cnd 11297 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
2625ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ)
27 1cnd 10000 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ)
2823, 26, 27addcan2d 10184 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((#‘𝑇) + 1) = ((𝑁 + 1) + 1) ↔ (#‘𝑇) = (𝑁 + 1)))
2916, 20, 283bitrd 294 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ (#‘𝑇) = (𝑁 + 1)))
30 opeq2 4371 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 + 1) = (#‘𝑇) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑇)⟩)
3130eqcoms 2629 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑇) = (𝑁 + 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑇)⟩)
3231oveq2d 6620 . . . . . . . . . . . . . . . . . 18 ((#‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩))
33 swrdccat1 13395 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩) = 𝑇)
3413, 33syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (#‘𝑇)⟩) = 𝑇)
3532, 34sylan9eqr 2677 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (#‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
3635ex 450 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
3729, 36sylbid 230 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
38373ad2antr1 1224 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((#‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
399, 38sylbid 230 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
4039imp 445 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
41 oveq1 6611 . . . . . . . . . . . . . . 15 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩))
4241eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
43423ad2ant2 1081 . . . . . . . . . . . . 13 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
4443ad2antlr 762 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
4540, 44mpbird 247 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
4645eleq1d 2683 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
4746biimpd 219 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
4847ex 450 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
4948com23 86 . . . . . . 7 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalksN 𝐺) → ((#‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
505, 49syld 47 . . . . . 6 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ((#‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
5150com13 88 . . . . 5 ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
52513ad2ant2 1081 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
533, 52mpcom 38 . . 3 (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
5453com12 32 . 2 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
551, 2wwlksnext 26657 . . . . . . . . . . 11 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
56 eleq1 2686 . . . . . . . . . . 11 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
5755, 56syl5ibrcom 237 . . . . . . . . . 10 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))
58573exp 1261 . . . . . . . . 9 (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑆𝑉 → ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
5958com23 86 . . . . . . . 8 (𝑇 ∈ (𝑁 WWalksN 𝐺) → ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (𝑆𝑉 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
6059com14 96 . . . . . . 7 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ({( lastS ‘𝑇), 𝑆} ∈ 𝐸 → (𝑆𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
6160imp 445 . . . . . 6 ((𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑆𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
62613adant1 1077 . . . . 5 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑆𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6362com12 32 . . . 4 (𝑆𝑉 → ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6463adantl 482 . . 3 ((𝑁 ∈ ℕ0𝑆𝑉) → ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6564imp 445 . 2 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))
6654, 65impbid 202 1 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {cpr 4150  cop 4154  cfv 5847  (class class class)co 6604  cc 9878  0cc0 9880  1c1 9881   + caddc 9883  0cn0 11236  ..^cfzo 12406  #chash 13057  Word cword 13230   lastS clsw 13231   ++ cconcat 13232  ⟨“cs1 13233   substr csubstr 13234  Vtxcvtx 25774  Edgcedg 25839   WWalksN cwwlksn 26587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-wwlks 26591  df-wwlksn 26592
This theorem is referenced by:  wwlksnextwrd  26661
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