Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clwwlksf1 Structured version   Visualization version   GIF version

Theorem clwwlksf1 41224
 Description: Lemma 3 for clwwlksbij 41227: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.)
Hypotheses
Ref Expression
clwwlksbij.d 𝐷 = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
clwwlksbij.f 𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
Assertion
Ref Expression
clwwlksf1 (𝑁 ∈ ℕ → 𝐹:𝐷1-1→(𝑁 ClWWalkSN 𝐺))
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑡,𝐷   𝑡,𝐺,𝑤   𝑡,𝑁
Allowed substitution hints:   𝐷(𝑤)   𝐹(𝑤,𝑡)

Proof of Theorem clwwlksf1
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwwlksbij.d . . 3 𝐷 = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
2 clwwlksbij.f . . 3 𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
31, 2clwwlksf 41222 . 2 (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalkSN 𝐺))
41, 2clwwlksfv 41223 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = (𝑥 substr ⟨0, 𝑁⟩))
51, 2clwwlksfv 41223 . . . . . 6 (𝑦𝐷 → (𝐹𝑦) = (𝑦 substr ⟨0, 𝑁⟩))
64, 5eqeqan12d 2626 . . . . 5 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
76adantl 481 . . . 4 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
8 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑥 → ( lastS ‘𝑤) = ( lastS ‘𝑥))
9 fveq1 6102 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
108, 9eqeq12d 2625 . . . . . . . 8 (𝑤 = 𝑥 → (( lastS ‘𝑤) = (𝑤‘0) ↔ ( lastS ‘𝑥) = (𝑥‘0)))
1110, 1elrab2 3333 . . . . . . 7 (𝑥𝐷 ↔ (𝑥 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑥) = (𝑥‘0)))
12 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑦 → ( lastS ‘𝑤) = ( lastS ‘𝑦))
13 fveq1 6102 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
1412, 13eqeq12d 2625 . . . . . . . 8 (𝑤 = 𝑦 → (( lastS ‘𝑤) = (𝑤‘0) ↔ ( lastS ‘𝑦) = (𝑦‘0)))
1514, 1elrab2 3333 . . . . . . 7 (𝑦𝐷 ↔ (𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)))
1611, 15anbi12i 729 . . . . . 6 ((𝑥𝐷𝑦𝐷) ↔ ((𝑥 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0))))
17 eqid 2610 . . . . . . . . . 10 (Vtx‘𝐺) = (Vtx‘𝐺)
18 eqid 2610 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
1917, 18wwlknp 41045 . . . . . . . . 9 (𝑥 ∈ (𝑁 WWalkSN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥𝑖), (𝑥‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2017, 18wwlknp 41045 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑁 WWalkSN 𝐺) → (𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
21 simprlr 799 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑥) = (𝑁 + 1))
22 simpllr 795 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑦) = (𝑁 + 1))
2321, 22eqtr4d 2647 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑥) = (#‘𝑦))
2423ad2antlr 759 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (#‘𝑥) = (#‘𝑦))
25 nncn 10905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26 ax-1cn 9873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 ∈ ℂ
27 pncan 10166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
2827eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 𝑁 = ((𝑁 + 1) − 1))
2925, 26, 28sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ → 𝑁 = ((𝑁 + 1) − 1))
30 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝑥) = (𝑁 + 1) → ((#‘𝑥) − 1) = ((𝑁 + 1) − 1))
3130eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝑥) = (𝑁 + 1) → ((𝑁 + 1) − 1) = ((#‘𝑥) − 1))
3229, 31sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → 𝑁 = ((#‘𝑥) − 1))
3332opeq2d 4347 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ⟨0, 𝑁⟩ = ⟨0, ((#‘𝑥) − 1)⟩)
3433oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑥 substr ⟨0, 𝑁⟩) = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩))
3533oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑦 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))
3634, 35eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . 24 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩)))
3736ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))))
3837ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))))
3938adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))))
4039impcom 445 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩)))
4140biimpa 500 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))
42 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → 𝑦 ∈ Word (Vtx‘𝐺))
43 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → 𝑥 ∈ Word (Vtx‘𝐺))
4442, 43anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)))
4544adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)))
46 nnnn0 11176 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
47 0nn0 11184 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ ℕ0
4846, 47jctil 558 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (0 ∈ ℕ0𝑁 ∈ ℕ0))
4948adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (0 ∈ ℕ0𝑁 ∈ ℕ0))
50 nnre 10904 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
5150lep1d 10834 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
52 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝑥) = (𝑁 + 1) → (𝑁 ≤ (#‘𝑥) ↔ 𝑁 ≤ (𝑁 + 1)))
5351, 52syl5ibr 235 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
5453ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
5554adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
5655impcom 445 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑁 ≤ (#‘𝑥))
57 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝑦) = (𝑁 + 1) → (𝑁 ≤ (#‘𝑦) ↔ 𝑁 ≤ (𝑁 + 1)))
5851, 57syl5ibr 235 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑦) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
5958ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
6059adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
6160impcom 445 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑁 ≤ (#‘𝑦))
62 swrdspsleq 13301 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑥) ∧ 𝑁 ≤ (#‘𝑦))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖)))
6345, 49, 56, 61, 62syl112anc 1322 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖)))
64 lbfzo0 12375 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
6564biimpri 217 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
6665adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 0 ∈ (0..^𝑁))
67 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 → (𝑥𝑖) = (𝑥‘0))
68 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 → (𝑦𝑖) = (𝑦‘0))
6967, 68eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 0 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
7069rspcv 3278 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ∈ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖) → (𝑥‘0) = (𝑦‘0)))
7166, 70syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖) → (𝑥‘0) = (𝑦‘0)))
7263, 71sylbid 229 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → (𝑥‘0) = (𝑦‘0)))
7372imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥‘0) = (𝑦‘0))
74 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ( lastS ‘𝑥) = (𝑥‘0))
75 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ( lastS ‘𝑦) = (𝑦‘0))
7674, 75eqeqan12rd 2628 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (( lastS ‘𝑥) = ( lastS ‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
7776ad2antlr 759 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (( lastS ‘𝑥) = ( lastS ‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
7873, 77mpbird 246 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → ( lastS ‘𝑥) = ( lastS ‘𝑦))
7924, 41, 78jca32 556 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → ((#‘𝑥) = (#‘𝑦) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ ( lastS ‘𝑥) = ( lastS ‘𝑦))))
8043adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → 𝑥 ∈ Word (Vtx‘𝐺))
8180adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑥 ∈ Word (Vtx‘𝐺))
8242adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → 𝑦 ∈ Word (Vtx‘𝐺))
8382adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑦 ∈ Word (Vtx‘𝐺))
84 1red 9934 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 1 ∈ ℝ)
85 nngt0 10926 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 < 𝑁)
86 0lt1 10429 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 < 1
8786a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 < 1)
8850, 84, 85, 87addgt0d 10481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
89 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝑥) = (𝑁 + 1) → (0 < (#‘𝑥) ↔ 0 < (𝑁 + 1)))
9088, 89syl5ibr 235 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
9190ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
9291adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
9392impcom 445 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 0 < (#‘𝑥))
9481, 83, 933jca 1235 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (#‘𝑥)))
9594adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (#‘𝑥)))
96 2swrd1eqwrdeq 13306 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (#‘𝑥)) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ ( lastS ‘𝑥) = ( lastS ‘𝑦)))))
9795, 96syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ ( lastS ‘𝑥) = ( lastS ‘𝑦)))))
9879, 97mpbird 246 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → 𝑥 = 𝑦)
9998exp31 628 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
10099expdcom 454 . . . . . . . . . . . . . . 15 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))))
101100ex 449 . . . . . . . . . . . . . 14 ((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
1021013adant3 1074 . . . . . . . . . . . . 13 ((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
10320, 102syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (𝑁 WWalkSN 𝐺) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
104103imp 444 . . . . . . . . . . 11 ((𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))))
105104expdcom 454 . . . . . . . . . 10 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
1061053adant3 1074 . . . . . . . . 9 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥𝑖), (𝑥‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
10719, 106syl 17 . . . . . . . 8 (𝑥 ∈ (𝑁 WWalkSN 𝐺) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
108107imp31 447 . . . . . . 7 (((𝑥 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0))) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
109108com12 32 . . . . . 6 (𝑁 ∈ ℕ → (((𝑥 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalkSN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
11016, 109syl5bi 231 . . . . 5 (𝑁 ∈ ℕ → ((𝑥𝐷𝑦𝐷) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
111110imp 444 . . . 4 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))
1127, 111sylbid 229 . . 3 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
113112ralrimivva 2954 . 2 (𝑁 ∈ ℕ → ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
114 dff13 6416 . 2 (𝐹:𝐷1-1→(𝑁 ClWWalkSN 𝐺) ↔ (𝐹:𝐷⟶(𝑁 ClWWalkSN 𝐺) ∧ ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1153, 113, 114sylanbrc 695 1 (𝑁 ∈ ℕ → 𝐹:𝐷1-1→(𝑁 ClWWalkSN 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150  Vtxcvtx 25673  Edgcedga 25792   WWalkSN cwwlksn 41029   ClWWalkSN cclwwlksn 41184 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-fal 1481  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-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-s1 13157  df-substr 13158  df-wwlks 41033  df-wwlksn 41034  df-clwwlks 41185  df-clwwlksn 41186 This theorem is referenced by:  clwwlksf1o  41226
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