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Theorem clwwlkf1 30068
Description: Lemma 3 for clwwlkf1o 30070: 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 30066 . 2 (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))
41, 2clwwlkfv 30067 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = (𝑥 prefix 𝑁))
51, 2clwwlkfv 30067 . . . . . 6 (𝑦𝐷 → (𝐹𝑦) = (𝑦 prefix 𝑁))
64, 5eqeqan12d 2751 . . . . 5 ((𝑥𝐷𝑦𝐷) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)))
76adantl 481 . . . 4 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)))
8 fveq2 6906 . . . . . . . . 9 (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
9 fveq1 6905 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
108, 9eqeq12d 2753 . . . . . . . 8 (𝑤 = 𝑥 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑥) = (𝑥‘0)))
1110, 1elrab2 3695 . . . . . . 7 (𝑥𝐷 ↔ (𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)))
12 fveq2 6906 . . . . . . . . 9 (𝑤 = 𝑦 → (lastS‘𝑤) = (lastS‘𝑦))
13 fveq1 6905 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
1412, 13eqeq12d 2753 . . . . . . . 8 (𝑤 = 𝑦 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑦) = (𝑦‘0)))
1514, 1elrab2 3695 . . . . . . 7 (𝑦𝐷 ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)))
1611, 15anbi12i 628 . . . . . 6 ((𝑥𝐷𝑦𝐷) ↔ ((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0))))
17 eqid 2737 . . . . . . . . . 10 (Vtx‘𝐺) = (Vtx‘𝐺)
18 eqid 2737 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
1917, 18wwlknp 29863 . . . . . . . . 9 (𝑥 ∈ (𝑁 WWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥𝑖), (𝑥‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2017, 18wwlknp 29863 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
21 simprlr 780 . . . . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . 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 12274 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26 ax-1cn 11213 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ ℂ
27 pncan 11514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
2827eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 𝑁 = ((𝑁 + 1) − 1))
2925, 26, 28sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ → 𝑁 = ((𝑁 + 1) − 1))
30 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((♯‘𝑥) = (𝑁 + 1) → ((♯‘𝑥) − 1) = ((𝑁 + 1) − 1))
3130eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑥) = (𝑁 + 1) → ((𝑁 + 1) − 1) = ((♯‘𝑥) − 1))
3229, 31sylan9eqr 2799 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → 𝑁 = ((♯‘𝑥) − 1))
3332oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑥 prefix 𝑁) = (𝑥 prefix ((♯‘𝑥) − 1)))
3432oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑦 prefix 𝑁) = (𝑦 prefix ((♯‘𝑥) − 1)))
3533, 34eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . 24 (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1))))
3635ex 412 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
3736ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . 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 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 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
3938impcom 407 . . . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → 𝑦 ∈ Word (Vtx‘𝐺))
42 simpll 767 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → 𝑥 ∈ Word (Vtx‘𝐺))
4341, 42anim12ci 614 . . . . . . . . . . . . . . . . . . . . . . . 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 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)))
45 nnnn0 12533 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
46 0nn0 12541 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ ℕ0
4745, 46jctil 519 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (0 ∈ ℕ0𝑁 ∈ ℕ0))
4847adantr 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))
49 nnre 12273 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
5049lep1d 12199 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
51 breq2 5147 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ≤ (♯‘𝑥) ↔ 𝑁 ≤ (𝑁 + 1)))
5250, 51imbitrrid 246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
5352ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
5453adantl 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)))
5554impcom 407 . . . . . . . . . . . . . . . . . . . . . . 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 5147 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑦) = (𝑁 + 1) → (𝑁 ≤ (♯‘𝑦) ↔ 𝑁 ≤ (𝑁 + 1)))
5750, 56imbitrrid 246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑦) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
5857ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
5958adantr 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)))
6059impcom 407 . . . . . . . . . . . . . . . . . . . . . . 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 14711 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑥 prefix 𝑁) = (𝑥 substr ⟨0, 𝑁⟩))
6261ad2ant2rl 749 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑥 prefix 𝑁) = (𝑥 substr ⟨0, 𝑁⟩))
63 pfxval 14711 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, 𝑁⟩))
6463ad2ant2l 746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, 𝑁⟩))
6562, 64eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . . . . . . . . . . . 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 14703 . . . . . . . . . . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . . . . . . . . . . . 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 1376 . . . . . . . . . . . . . . . . . . . . . 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 13739 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
7170biimpri 228 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
7271adantr 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))
73 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 → (𝑥𝑖) = (𝑥‘0))
74 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 → (𝑦𝑖) = (𝑦‘0))
7573, 74eqeq12d 2753 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 0 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
7675rspcv 3618 . . . . . . . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (lastS‘𝑥) = (𝑥‘0))
81 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (lastS‘𝑦) = (𝑦‘0))
8280, 81eqeqan12rd 2752 . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . 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 515 . . . . . . . . . . . . . . . . . 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 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))
8786adantl 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))
8841adantr 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))
8988adantl 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))
90 1red 11262 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 1 ∈ ℝ)
91 nngt0 12297 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 < 𝑁)
92 0lt1 11785 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 < 1
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 < 1)
9449, 90, 91, 93addgt0d 11838 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
95 breq2 5147 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((♯‘𝑥) = (𝑁 + 1) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 1)))
9694, 95imbitrrid 246 . . . . . . . . . . . . . . . . . . . . . . . 24 ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
9796ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
9897adantl 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)))
9998impcom 407 . . . . . . . . . . . . . . . . . . . . 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 1129 . . . . . . . . . . . . . . . . . . . 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 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 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (♯‘𝑥)))
102 pfxsuff1eqwrdeq 14737 . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . 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 406 . . . . . . . . . . 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 414 . . . . . . . . . 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 1133 . . . . . . . . 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 417 . . . . . . 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, 115biimtrid 242 . . . . 5 (𝑁 ∈ ℕ → ((𝑥𝐷𝑦𝐷) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
117116imp 406 . . . 4 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦))
1187, 117sylbid 240 . . 3 ((𝑁 ∈ ℕ ∧ (𝑥𝐷𝑦𝐷)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
119118ralrimivva 3202 . 2 (𝑁 ∈ ℕ → ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
120 dff13 7275 . 2 (𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1213, 119, 120sylanbrc 583 1 (𝑁 ∈ ℕ → 𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  {cpr 4628  cop 4632   class class class wbr 5143  cmpt 5225  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  cc 11153  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  cn 12266  0cn0 12526  ..^cfzo 13694  chash 14369  Word cword 14552  lastSclsw 14600   substr csubstr 14678   prefix cpfx 14708  Vtxcvtx 29013  Edgcedg 29064   WWalksN cwwlksn 29846   ClWWalksN cclwwlkn 30043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-lsw 14601  df-s1 14634  df-substr 14679  df-pfx 14709  df-wwlks 29850  df-wwlksn 29851  df-clwwlk 30001  df-clwwlkn 30044
This theorem is referenced by:  clwwlkf1o  30070
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