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Theorem wwlksnredwwlkn0 27010
 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 27009 . . . 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 474 . . . . . . . . 9 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦)
54adantl 473 . . . . . . . 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 6347 . . . . . . . . . . . . . 14 (𝑦 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
76eqcoms 2764 . . . . . . . . . . . . 13 ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → (𝑦‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
87adantr 472 . . . . . . . . . . . 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 2756 . . . . . . . . . . . . . . . . . . 19 (Vtx‘𝐺) = (Vtx‘𝐺)
109, 1wwlknp 26942 . . . . . . . . . . . . . . . . . 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 11520 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
12 peano2nn0 11521 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
13 nn0re 11489 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℕ0 → (𝑁 + 1) ∈ ℝ)
14 lep1 11050 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℝ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
1512, 13, 143syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
16 peano2nn0 11521 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℕ0)
1716nn0zd 11668 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) ∈ ℕ0 → ((𝑁 + 1) + 1) ∈ ℤ)
18 fznn 12597 . . . . . . . . . . . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
21 oveq2 6817 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘𝑊) = ((𝑁 + 1) + 1) → (1...(♯‘𝑊)) = (1...((𝑁 + 1) + 1)))
2221eleq2d 2821 . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (1...(♯‘𝑊))))
25 simpl 474 . . . . . . . . . . . . . . . . . . . 20 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → 𝑊 ∈ Word (Vtx‘𝐺))
2624, 25jctild 567 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ0 → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊)))))
27263adant3 1127 . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 15 (((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
3130adantr 472 . . . . . . . . . . . . . 14 ((((𝑊‘0) = 𝑃 ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) ∧ 𝑦 ∈ (𝑁 WWalksN 𝐺)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
3231adantl 473 . . . . . . . . . . . . 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 13636 . . . . . . . . . . . . 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 821 . . . . . . . . . . . 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 2794 . . . . . . . . . . 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 472 . . . . . . . . 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 479 . . . . . . . . 9 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) → {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)
4140adantl 473 . . . . . . . 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 1123 . . . . . . 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 3151 . . . . 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 2762 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
5049adantl 473 . . . . . . 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 6347 . . . . . . . . 9 ((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑦‘0))
5251adantr 472 . . . . . . . 8 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑦‘0))
5352adantr 472 . . . . . . 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 479 . . . . . . . 8 (((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (𝑦‘0) = 𝑃)
5554adantr 472 . . . . . . 7 ((((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) ∧ (𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺))) → (𝑦‘0) = 𝑃)
5650, 53, 553eqtrd 2794 . . . . . 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 1127 . . . 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 3168 . 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 1072   = wceq 1628   ∈ wcel 2135  ∀wral 3046  ∃wrex 3047  {cpr 4319  ⟨cop 4323   class class class wbr 4800  ‘cfv 6045  (class class class)co 6809  ℝcr 10123  0cc0 10124  1c1 10125   + caddc 10127   ≤ cle 10263  ℕcn 11208  ℕ0cn0 11480  ℤcz 11565  ...cfz 12515  ..^cfzo 12655  ♯chash 13307  Word cword 13473  lastSclsw 13474   substr csubstr 13477  Vtxcvtx 26069  Edgcedg 26134   WWalksN cwwlksn 26925 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-map 8021  df-pm 8022  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-card 8951  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-n0 11481  df-z 11566  df-uz 11876  df-fz 12516  df-fzo 12656  df-hash 13308  df-word 13481  df-lsw 13482  df-substr 13485  df-wwlks 26929  df-wwlksn 26930 This theorem is referenced by:  rusgrnumwwlks  27092
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