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Theorem wwlksnextbi 29914
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.) (Proof shortened by AV, 27-Oct-2022.)
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 29863 . . . 4 (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
4 wwlksnred 29912 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
54ad2antrr 726 . . . . . . 7 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
6 fveqeq2 6915 . . . . . . . . . . . . . . . 16 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
763ad2ant2 1135 . . . . . . . . . . . . . . 15 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
87adantl 481 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) ↔ (♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
9 s1cl 14640 . . . . . . . . . . . . . . . . . . . . 21 (𝑆𝑉 → ⟨“𝑆”⟩ ∈ Word 𝑉)
109adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝑆𝑉) → ⟨“𝑆”⟩ ∈ Word 𝑉)
1110anim1ci 616 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉))
12 ccatlen 14613 . . . . . . . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1)))
15 s1len 14644 . . . . . . . . . . . . . . . . . . . 20 (♯‘⟨“𝑆”⟩) = 1
1615a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘⟨“𝑆”⟩) = 1)
1716oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((♯‘𝑇) + 1))
1817eqeq1d 2739 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + (♯‘⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ ((♯‘𝑇) + 1) = ((𝑁 + 1) + 1)))
19 lencl 14571 . . . . . . . . . . . . . . . . . . . 20 (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℕ0)
2019nn0cnd 12589 . . . . . . . . . . . . . . . . . . 19 (𝑇 ∈ Word 𝑉 → (♯‘𝑇) ∈ ℂ)
2120adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (♯‘𝑇) ∈ ℂ)
22 peano2nn0 12566 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
2322nn0cnd 12589 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ)
2423ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (𝑁 + 1) ∈ ℂ)
25 1cnd 11256 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → 1 ∈ ℂ)
2621, 24, 25addcan2d 11465 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → (((♯‘𝑇) + 1) = ((𝑁 + 1) + 1) ↔ (♯‘𝑇) = (𝑁 + 1)))
2714, 18, 263bitrd 305 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) ↔ (♯‘𝑇) = (𝑁 + 1)))
28 oveq2 7439 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) = (♯‘𝑇) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)))
2928eqcoms 2745 . . . . . . . . . . . . . . . . . 18 ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)))
30 pfxccat1 14740 . . . . . . . . . . . . . . . . . . 19 ((𝑇 ∈ Word 𝑉 ∧ ⟨“𝑆”⟩ ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)) = 𝑇)
3111, 30syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (♯‘𝑇)) = 𝑇)
3229, 31sylan9eqr 2799 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) ∧ (♯‘𝑇) = (𝑁 + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇)
3332ex 412 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘𝑇) = (𝑁 + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
3427, 33sylbid 240 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ 𝑇 ∈ Word 𝑉) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
35343ad2antr1 1189 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘(𝑇 ++ ⟨“𝑆”⟩)) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
368, 35sylbid 240 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
3736imp 406 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇)
38 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 prefix (𝑁 + 1)) = ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)))
3938eqeq1d 2739 . . . . . . . . . . . . . 14 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
40393ad2ant2 1135 . . . . . . . . . . . . 13 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
4140ad2antlr 727 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) = 𝑇 ↔ ((𝑇 ++ ⟨“𝑆”⟩) prefix (𝑁 + 1)) = 𝑇))
4237, 41mpbird 257 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 prefix (𝑁 + 1)) = 𝑇)
4342eleq1d 2826 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
4443biimpd 229 . . . . . . . . 9 ((((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
4544ex 412 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
4645com23 86 . . . . . . 7 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → ((𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → 𝑇 ∈ (𝑁 WWalksN 𝐺))))
475, 46syld 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))))
4847com13 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))))
49483ad2ant2 1135 . . . 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))))
503, 49mpcom 38 . . 3 (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
5150com12 32 . 2 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → 𝑇 ∈ (𝑁 WWalksN 𝐺)))
521, 2wwlksnext 29913 . . . . . . . . . . 11 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
53 eleq1 2829 . . . . . . . . . . 11 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
5452, 53syl5ibrcom 247 . . . . . . . . . 10 ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))
55543exp 1120 . . . . . . . . 9 (𝑇 ∈ (𝑁 WWalksN 𝐺) → (𝑆𝑉 → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
5655com23 86 . . . . . . . 8 (𝑇 ∈ (𝑁 WWalksN 𝐺) → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑆𝑉 → (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
5756com14 96 . . . . . . 7 (𝑊 = (𝑇 ++ ⟨“𝑆”⟩) → ({(lastS‘𝑇), 𝑆} ∈ 𝐸 → (𝑆𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))))
5857imp 406 . . . . . 6 ((𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑆𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
59583adant1 1131 . . . . 5 ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑆𝑉 → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6059com12 32 . . . 4 (𝑆𝑉 → ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6160adantl 481 . . 3 ((𝑁 ∈ ℕ0𝑆𝑉) → ((𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))))
6261imp 406 . 2 (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑇 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)))
6351, 62impbid 212 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {cpr 4628  cfv 6561  (class class class)co 7431  cc 11153  0cc0 11155  1c1 11156   + caddc 11158  0cn0 12526  ..^cfzo 13694  chash 14369  Word cword 14552  lastSclsw 14600   ++ cconcat 14608  ⟨“cs1 14633   prefix cpfx 14708  Vtxcvtx 29013  Edgcedg 29064   WWalksN cwwlksn 29846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-wwlks 29850  df-wwlksn 29851
This theorem is referenced by:  wwlksnextwrd  29917
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