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Theorem wwlksnredwwlkn0 26859
 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 26858 . . . 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 6228 . . . . . . . . . . . . . 14 (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
76eqcoms 2659 . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . . . . . . 19 (Vtx‘𝐺) = (Vtx‘𝐺)
109, 1wwlknp 26791 . . . . . . . . . . . . . . . . . 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 11370 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
12 peano2nn0 11371 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
13 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℕ0 → (𝑁 + 1) ∈ ℝ)
14 lep1 10900 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℝ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
1512, 13, 143syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
16 peano2nn0 11371 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ0)
1716nn0zd 11518 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℤ)
18 fznn 12446 . . . . . . . . . . . . . . . . . . . . . . . 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 977 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
21 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑊) = ((𝑁 + 1) + 1) → (1...(#‘𝑊)) = (1...((𝑁 + 1) + 1)))
2221eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (1...(#‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
2320, 22syl5ibr 236 . . . . . . . . . . . . . . . . . . . . 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 565 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(#‘𝑊)))))
27263adant3 1101 . . . . . . . . . . . . . . . . . 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 13486 . . . . . . . . . . . . 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 819 . . . . . . . . . . . 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 2689 . . . . . . . . . . 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 1261 . . . . . . 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 3046 . . . . 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 88 . . 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 38 . 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 2657 . . . . . . . 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 6228 . . . . . . . . 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 2689 . . . . . 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 1101 . . . 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 3063 . 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 202 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  {cpr 4212  ⟨cop 4216   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   ≤ cle 10113  ℕcn 11058  ℕ0cn0 11330  ℤcz 11415  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323   lastS clsw 13324   substr csubstr 13327  Vtxcvtx 25919  Edgcedg 25984   WWalksN cwwlksn 26774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-lsw 13332  df-substr 13335  df-wwlks 26778  df-wwlksn 26779 This theorem is referenced by:  rusgrnumwwlks  26941
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