Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wwlksnredwwlkn0 Structured version   Visualization version   GIF version

Theorem wwlksnredwwlkn0 41102
 Description: For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.)
Hypothesis
Ref Expression
wwlksnredwwlkn.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
wwlksnredwwlkn0 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)))
Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑁   𝑦,𝑊   𝑦,𝑃

Proof of Theorem wwlksnredwwlkn0
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wwlksnredwwlkn.e . . . . 5 𝐸 = (Edg‘𝐺)
21wwlksnredwwlkn 41101 . . . 4 (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) → ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)))
32imp 444 . . 3 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸))
4 simpl 472 . . . . . . . . 9 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦)
54adantl 481 . . . . . . . 8 (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺)) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦)
6 fveq1 6102 . . . . . . . . . . . . . 14 (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
76eqcoms 2618 . . . . . . . . . . . . 13 ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
87adantr 480 . . . . . . . . . . . 12 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺))) → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
9 eqid 2610 . . . . . . . . . . . . . . . . . . 19 (Vtx‘𝐺) = (Vtx‘𝐺)
109, 1wwlknp 41045 . . . . . . . . . . . . . . . . . 18 (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
11 nn0p1nn 11209 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
12 peano2nn0 11210 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
13 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℕ0 → (𝑁 + 1) ∈ ℝ)
14 lep1 10741 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℝ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
1512, 13, 143syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
16 peano2nn0 11210 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ0)
1716nn0zd 11356 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℤ)
18 fznn 12278 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑁 + 1) + 1) ∈ ℤ → ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ ℕ ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1))))
1912, 17, 183syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → ((𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ ℕ ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1))))
2011, 15, 19mpbir2and 959 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
21 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑊) = ((𝑁 + 1) + 1) → (1...(#‘𝑊)) = (1...((𝑁 + 1) + 1)))
2221eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
2320, 22syl5ibr 235 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(#‘𝑊))))
2423adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(#‘𝑊))))
25 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → 𝑊 ∈ Word (Vtx‘𝐺))
2624, 25jctild 564 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))))
27263adant3 1074 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))))
2810, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))))
2928impcom 445 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
3029adantl 481 . . . . . . . . . . . . . . 15 (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
3130adantr 480 . . . . . . . . . . . . . 14 ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
3231adantl 481 . . . . . . . . . . . . 13 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))))
33 swrd0fv0 13292 . . . . . . . . . . . . 13 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))
3432, 33syl 17 . . . . . . . . . . . 12 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))
35 simprll 798 . . . . . . . . . . . 12 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺))) → (𝑊‘0) = 𝑃)
368, 34, 353eqtrd 2648 . . . . . . . . . . 11 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺))) → (𝑦‘0) = 𝑃)
3736ex 449 . . . . . . . . . 10 ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺)) → (𝑦‘0) = 𝑃))
3837adantr 480 . . . . . . . . 9 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺)) → (𝑦‘0) = 𝑃))
3938impcom 445 . . . . . . . 8 (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺)) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)) → (𝑦‘0) = 𝑃)
40 simpr 476 . . . . . . . . 9 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)
4140adantl 481 . . . . . . . 8 (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺)) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)) → {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)
425, 39, 413jca 1235 . . . . . . 7 (((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺)) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸))
4342ex 449 . . . . . 6 ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalkSN 𝐺)) → (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)))
4443reximdva 3000 . . . . 5 (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) → (∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)))
4544ex 449 . . . 4 ((𝑊‘0) = 𝑃 → ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸))))
4645com13 86 . . 3 (∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → ((𝑊‘0) = 𝑃 → ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸))))
473, 46mpcom 37 . 2 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → ((𝑊‘0) = 𝑃 → ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)))
4829, 33syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))
4948eqcomd 2616 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (𝑊‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
5049adantl 481 . . . . . . 7 ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) → (𝑊‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
51 fveq1 6102 . . . . . . . . 9 ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑦‘0))
5251adantr 480 . . . . . . . 8 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑦‘0))
5352adantr 480 . . . . . . 7 ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑦‘0))
54 simpr 476 . . . . . . . 8 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (𝑦‘0) = 𝑃)
5554adantr 480 . . . . . . 7 ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) → (𝑦‘0) = 𝑃)
5650, 53, 553eqtrd 2648 . . . . . 6 ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺))) → (𝑊‘0) = 𝑃)
5756ex 449 . . . . 5 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (𝑊‘0) = 𝑃))
58573adant3 1074 . . . 4 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (𝑊‘0) = 𝑃))
5958com12 32 . . 3 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → (𝑊‘0) = 𝑃))
6059rexlimdvw 3016 . 2 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → (∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸) → (𝑊‘0) = 𝑃))
6147, 60impbid 201 1 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalkSN 𝐺)) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalkSN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150  Vtxcvtx 25673  Edgcedga 25792   WWalkSN cwwlksn 41029 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-substr 13158  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by:  rusgrnumwwlks  41177
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