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Theorem clwwlkf1 27813
Description: Lemma 3 for clwwlkf1o 27815: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
clwwlkf1o.d 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
clwwlkf1o.f 𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))
Assertion
Ref Expression
clwwlkf1 (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 clwwlkf1
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwwlkf1o.d . . 3 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
2 clwwlkf1o.f . . 3 𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))
31, 2clwwlkf 27811 . 2 (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))
41, 2clwwlkfv 27812 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = (𝑥 prefix 𝑁))
51, 2clwwlkfv 27812 . . . . . 6 (𝑦𝐷 → (𝐹𝑦) = (𝑦 prefix 𝑁))
64, 5eqeqan12d 2838 . . . . 5 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)))
76adantl 485 . . . 4 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)))
8 fveq2 6643 . . . . . . . . 9 (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
9 fveq1 6642 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
108, 9eqeq12d 2837 . . . . . . . 8 (𝑤 = 𝑥 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑥) = (𝑥‘0)))
1110, 1elrab2 3660 . . . . . . 7 (𝑥𝐷 ↔ (𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)))
12 fveq2 6643 . . . . . . . . 9 (𝑤 = 𝑦 → (lastS‘𝑤) = (lastS‘𝑦))
13 fveq1 6642 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
1412, 13eqeq12d 2837 . . . . . . . 8 (𝑤 = 𝑦 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑦) = (𝑦‘0)))
1514, 1elrab2 3660 . . . . . . 7 (𝑦𝐷 ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)))
1611, 15anbi12i 629 . . . . . 6 ((𝑥𝐷𝑦𝐷) ↔ ((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0))))
17 eqid 2821 . . . . . . . . . 10 (Vtx‘𝐺) = (Vtx‘𝐺)
18 eqid 2821 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
1917, 18wwlknp 27608 . . . . . . . . 9 (𝑥 ∈ (𝑁 WWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥𝑖), (𝑥‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2017, 18wwlknp 27608 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
21 simprlr 779 . . . . . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → (♯‘𝑥) = (♯‘𝑦))
25 nncn 11623 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26 ax-1cn 10572 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ ℂ
27 pncan 10869 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
2827eqcomd 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 𝑁 = ((𝑁 + 1) − 1))
2925, 26, 28sylancl 589 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ → 𝑁 = ((𝑁 + 1) − 1))
30 oveq1 7137 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝑥) = (𝑁 + 1) → ((♯‘𝑥) − 1) = ((𝑁 + 1) − 1))
3130eqcomd 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑥) = (𝑁 + 1) → ((𝑁 + 1) − 1) = ((♯‘𝑥) − 1))
3229, 31sylan9eqr 2878 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → 𝑁 = ((♯‘𝑥) − 1))
3332oveq2d 7146 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑥 prefix 𝑁) = (𝑥 prefix ((♯‘𝑥) − 1)))
3432oveq2d 7146 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑦 prefix 𝑁) = (𝑦 prefix ((♯‘𝑥) − 1)))
3533, 34eqeq12d 2837 . . . . . . . . . . . . . . . . . . . . . . . 24 (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1))))
3635ex 416 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
3736ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
3837adantl 485 . . . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
3938impcom 411 . . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1))))
4039biimpa 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))
41 simpll 766 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → 𝑦 ∈ Word (Vtx‘𝐺))
42 simpll 766 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → 𝑥 ∈ Word (Vtx‘𝐺))
4341, 42anim12ci 616 . . . . . . . . . . . . . . . . . . . . . . . 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)))
4443adantl 485 . . . . . . . . . . . . . . . . . . . . . . 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)))
45 nnnn0 11882 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
46 0nn0 11890 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ ℕ0
4745, 46jctil 523 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (0 ∈ ℕ0𝑁 ∈ ℕ0))
4847adantr 484 . . . . . . . . . . . . . . . . . . . . . . 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))
49 nnre 11622 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
5049lep1d 11548 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
51 breq2 5043 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ≤ (♯‘𝑥) ↔ 𝑁 ≤ (𝑁 + 1)))
5250, 51syl5ibr 249 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
5352ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
5453adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 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)))
5554impcom 411 . . . . . . . . . . . . . . . . . . . . . . 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))
56 breq2 5043 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑦) = (𝑁 + 1) → (𝑁 ≤ (♯‘𝑦) ↔ 𝑁 ≤ (𝑁 + 1)))
5750, 56syl5ibr 249 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑦) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
5857ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
5958adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 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)))
6059impcom 411 . . . . . . . . . . . . . . . . . . . . . . 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))
61 pfxval 14014 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑥 prefix 𝑁) = (𝑥 substr ⟨0, 𝑁⟩))
6261ad2ant2rl 748 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑥 prefix 𝑁) = (𝑥 substr ⟨0, 𝑁⟩))
63 pfxval 14014 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, 𝑁⟩))
6463ad2ant2l 745 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, 𝑁⟩))
6562, 64eqeq12d 2837 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0)) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
66653adant3 1129 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (♯‘𝑥) ∧ 𝑁 ≤ (♯‘𝑦))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
67 swrdspsleq 14006 . . . . . . . . . . . . . . . . . . . . . . . 24 (((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)))
6866, 67bitrd 282 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (♯‘𝑥) ∧ 𝑁 ≤ (♯‘𝑦))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖)))
6944, 48, 55, 60, 68syl112anc 1371 . . . . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖)))
70 lbfzo0 13060 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
7170biimpri 231 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
7271adantr 484 . . . . . . . . . . . . . . . . . . . . . . 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))
73 fveq2 6643 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 → (𝑥𝑖) = (𝑥‘0))
74 fveq2 6643 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 → (𝑦𝑖) = (𝑦‘0))
7573, 74eqeq12d 2837 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 0 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
7675rspcv 3595 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ∈ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖) → (𝑥‘0) = (𝑦‘0)))
7772, 76syl 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)))
7869, 77sylbid 243 . . . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) → (𝑥‘0) = (𝑦‘0)))
7978imp 410 . . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → (𝑥‘0) = (𝑦‘0))
80 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (lastS‘𝑥) = (𝑥‘0))
81 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (lastS‘𝑦) = (𝑦‘0))
8280, 81eqeqan12rd 2840 . . . . . . . . . . . . . . . . . . . . 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)))
8382ad2antlr 726 . . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → ((lastS‘𝑥) = (lastS‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
8479, 83mpbird 260 . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → (lastS‘𝑥) = (lastS‘𝑦))
8524, 40, 84jca32 519 . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → ((♯‘𝑥) = (♯‘𝑦) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)) ∧ (lastS‘𝑥) = (lastS‘𝑦))))
8642adantl 485 . . . . . . . . . . . . . . . . . . . . . 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))
8786adantl 485 . . . . . . . . . . . . . . . . . . . . 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))
8841adantr 484 . . . . . . . . . . . . . . . . . . . . . 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))
8988adantl 485 . . . . . . . . . . . . . . . . . . . . 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))
90 1red 10619 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 1 ∈ ℝ)
91 nngt0 11646 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 < 𝑁)
92 0lt1 11139 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 < 1
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 < 1)
9449, 90, 91, 93addgt0d 11192 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
95 breq2 5043 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘𝑥) = (𝑁 + 1) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 1)))
9694, 95syl5ibr 249 . . . . . . . . . . . . . . . . . . . . . . . 24 ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
9796ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
9897adantl 485 . . . . . . . . . . . . . . . . . . . . . 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)))
9998impcom 411 . . . . . . . . . . . . . . . . . . . . 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))
10087, 89, 993jca 1125 . . . . . . . . . . . . . . . . . . . 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)))
101100adantr 484 . . . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (♯‘𝑥)))
102 pfxsuff1eqwrdeq 14040 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (♯‘𝑥)) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)) ∧ (lastS‘𝑥) = (lastS‘𝑦)))))
103101, 102syl 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)) ∧ (lastS‘𝑥) = (lastS‘𝑦)))))
10485, 103mpbird 260 . . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁)) → 𝑥 = 𝑦)
105104exp31 423 . . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
106105expdcom 418 . . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦))))
107106ex 416 . . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
1081073adant3 1129 . . . . . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
10920, 108syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (𝑁 WWalksN 𝐺) → ((lastS‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
110109imp 410 . . . . . . . . . . 11 ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦))))
111110expdcom 418 . . . . . . . . . 10 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) → ((lastS‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
1121113adant3 1129 . . . . . . . . 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 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
11319, 112syl 17 . . . . . . . 8 (𝑥 ∈ (𝑁 WWalksN 𝐺) → ((lastS‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
114113imp31 421 . . . . . . 7 (((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0))) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
115114com12 32 . . . . . 6 (𝑁 ∈ ℕ → (((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
11616, 115syl5bi 245 . . . . 5 (𝑁 ∈ ℕ → ((𝑥𝐷𝑦𝐷) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
117116imp 410 . . . 4 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦))
1187, 117sylbid 243 . . 3 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
119118ralrimivva 3179 . 2 (𝑁 ∈ ℕ → ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
120 dff13 6987 . 2 (𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1213, 119, 120sylanbrc 586 1 (𝑁 ∈ ℕ → 𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  {crab 3130  {cpr 4542  cop 4546   class class class wbr 5039  cmpt 5119  wf 6324  1-1wf1 6325  cfv 6328  (class class class)co 7130  cc 10512  0cc0 10514  1c1 10515   + caddc 10517   < clt 10652  cle 10653  cmin 10847  cn 11615  0cn0 11875  ..^cfzo 13016  chash 13674  Word cword 13845  lastSclsw 13893   substr csubstr 13981   prefix cpfx 14011  Vtxcvtx 26768  Edgcedg 26819   WWalksN cwwlksn 27591   ClWWalksN cclwwlkn 27788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-n0 11876  df-xnn0 11946  df-z 11960  df-uz 12222  df-fz 12876  df-fzo 13017  df-hash 13675  df-word 13846  df-lsw 13894  df-s1 13929  df-substr 13982  df-pfx 14012  df-wwlks 27595  df-wwlksn 27596  df-clwwlk 27746  df-clwwlkn 27789
This theorem is referenced by:  clwwlkf1o  27815
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