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Theorem wwlksnextbi 26933
 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 26867 . . . 4 (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
4 wwlksnred 26931 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalksN 𝐺)))
54ad2antrr 764 . . . . . . 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 6304 . . . . . . . . . . . . . . . . 17 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (♯‘𝑊) = (♯‘(𝑇 ++ ⟨“𝑆”⟩)))
76eqeq1d 2726 . . . . . . . . . . . . . . . 16 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
873ad2ant2 1126 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
98adantl 473 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
10 s1cl 13493 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
1110adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0𝑆𝑉) → ⟨“𝑆”⟩ ∈ Word 𝑉)
1211anim2i 594 . . . . . . . . . . . . . . . . . . . 20 ((𝑇 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0𝑆𝑉)) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
1312ancoms 468 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
14 ccatlen 13468 . . . . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
17 s1len 13497 . . . . . . . . . . . . . . . . . . . 20 (♯‘⟨“𝑆”⟩) = 1
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘⟨“𝑆”⟩) = 1)
1918oveq2d 6781 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((♯‘𝑇) + 1))
2019eqeq1d 2726 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) + 1) = ((𝑁 + 1) + 1)))
21 lencl 13431 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℕ0)
2221nn0cnd 11466 . . . . . . . . . . . . . . . . . . 19 (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℂ)
2322adantl 473 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘𝑇) ∈ ℂ)
24 peano2nn0 11446 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
2524nn0cnd 11466 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
2625ad2antrr 764 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ)
27 1cnd 10169 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ)
2823, 26, 27addcan2d 10353 . . . . . . . . . . . . . . . . 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 4510 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 + 1) = (♯‘𝑇) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (♯‘𝑇)⟩)
3130eqcoms 2732 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑇) = (𝑁 + 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (♯‘𝑇)⟩)
3231oveq2d 6781 . . . . . . . . . . . . . . . . . 18 ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (♯‘𝑇)⟩))
33 swrdccat1 13578 . . . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (♯‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇)
3635ex 449 . . . . . . . . . . . . . . . 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 1180 . . . . . . . . . . . . . 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 444 . . . . . . . . . . . 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 6772 . . . . . . . . . . . . . . 15 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩))
4241eqeq1d 2726 . . . . . . . . . . . . . 14 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
43423ad2ant2 1126 . . . . . . . . . . . . 13 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑇))
4443ad2antlr 765 . . . . . . . . . . . 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 2788 . . . . . . . . . 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 449 . . . . . . . 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 1126 . . . 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 26932 . . . . . . . . . . 11 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
56 eleq1 2791 . . . . . . . . . . 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 1112 . . . . . . . . 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 444 . . . . . 6 ((𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑆𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
62613adant1 1122 . . . . 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 473 . . 3 ((𝑁 ∈ ℕ0𝑆𝑉) → ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6564imp 444 . 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 383   ∧ w3a 1072   = wceq 1596   ∈ wcel 2103  ∀wral 3014  {cpr 4287  ⟨cop 4291  ‘cfv 6001  (class class class)co 6765  ℂcc 10047  0cc0 10049  1c1 10050   + caddc 10052  ℕ0cn0 11405  ..^cfzo 12580  ♯chash 13232  Word cword 13398   lastS clsw 13399   ++ cconcat 13400  ⟨“cs1 13401   substr csubstr 13402  Vtxcvtx 25994  Edgcedg 26059   WWalksN cwwlksn 26850 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581  df-hash 13233  df-word 13406  df-lsw 13407  df-concat 13408  df-s1 13409  df-substr 13410  df-wwlks 26854  df-wwlksn 26855 This theorem is referenced by:  wwlksnextwrd  26936
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