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Theorem clwwlkf1 26324
 Description: Lemma 3 for clwwlkbij 26327: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)
Hypotheses
Ref Expression
clwwlkbij.d 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
clwwlkbij.f 𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
Assertion
Ref Expression
clwwlkf1 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑉   𝑡,𝐷   𝑡,𝐸,𝑤   𝑡,𝑁   𝑡,𝑉   𝑡,𝑋   𝑡,𝑌
Allowed substitution hints:   𝐷(𝑤)   𝐹(𝑤,𝑡)   𝑋(𝑤)   𝑌(𝑤)

Proof of Theorem clwwlkf1
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwwlkbij.d . . 3 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
2 clwwlkbij.f . . 3 𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
31, 2clwwlkf 26322 . 2 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁))
41, 2clwwlkfv 26323 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = (𝑥 substr ⟨0, 𝑁⟩))
51, 2clwwlkfv 26323 . . . . . 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 (((𝑉𝑋𝐸𝑌𝑁 ∈ 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 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( 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 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)))
1611, 15anbi12i 729 . . . . . 6 ((𝑥𝐷𝑦𝐷) ↔ ((𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0))))
17 wwlknimp 26215 . . . . . . . . 9 (𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥𝑖), (𝑥‘(𝑖 + 1))} ∈ ran 𝐸))
18 wwlknimp 26215 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ ran 𝐸))
19 simprlr 799 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑥) = (𝑁 + 1))
20 simpllr 795 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑦) = (𝑁 + 1))
2119, 20eqtr4d 2647 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑥) = (#‘𝑦))
2221ad2antlr 759 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (#‘𝑥) = (#‘𝑦))
23 nncn 10905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
24 ax-1cn 9873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1 ∈ ℂ
25 pncan 10166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
2625eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 𝑁 = ((𝑁 + 1) − 1))
2723, 24, 26sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ ℕ → 𝑁 = ((𝑁 + 1) − 1))
28 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝑥) = (𝑁 + 1) → ((#‘𝑥) − 1) = ((𝑁 + 1) − 1))
2928eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝑥) = (𝑁 + 1) → ((𝑁 + 1) − 1) = ((#‘𝑥) − 1))
3027, 29sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → 𝑁 = ((#‘𝑥) − 1))
3130opeq2d 4347 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ⟨0, 𝑁⟩ = ⟨0, ((#‘𝑥) − 1)⟩)
3231oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑥 substr ⟨0, 𝑁⟩) = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩))
3331oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑦 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))
3432, 33eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩)))
3534ex 449 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))))
3635ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ Word 𝑉 ∧ (#‘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)⟩))))
3736adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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)⟩))))
3837impcom 445 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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)⟩)))
3938biimpa 500 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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)⟩))
40 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → 𝑦 ∈ Word 𝑉)
41 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → 𝑥 ∈ Word 𝑉)
4240, 41anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
4342adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
44 nnnn0 11176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
45 0nn0 11184 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ∈ ℕ0
4644, 45jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (0 ∈ ℕ0𝑁 ∈ ℕ0))
4746adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (0 ∈ ℕ0𝑁 ∈ ℕ0))
48 nnre 10904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
4948lep1d 10834 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
50 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝑥) = (𝑁 + 1) → (𝑁 ≤ (#‘𝑥) ↔ 𝑁 ≤ (𝑁 + 1)))
5149, 50syl5ibr 235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
5251ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
5352adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
5453impcom 445 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑁 ≤ (#‘𝑥))
55 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝑦) = (𝑁 + 1) → (𝑁 ≤ (#‘𝑦) ↔ 𝑁 ≤ (𝑁 + 1)))
5649, 55syl5ibr 235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝑦) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
5756ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
5857adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
5958impcom 445 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑁 ≤ (#‘𝑦))
60 swrdspsleq 13301 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉) ∧ (0 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑥) ∧ 𝑁 ≤ (#‘𝑦))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖)))
6143, 47, 54, 59, 60syl112anc 1322 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖)))
62 lbfzo0 12375 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
6362biimpri 217 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
6463adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 0 ∈ (0..^𝑁))
65 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 = 0 → (𝑥𝑖) = (𝑥‘0))
66 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 = 0 → (𝑦𝑖) = (𝑦‘0))
6765, 66eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 0 → ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
6867rspcv 3278 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖) → (𝑥‘0) = (𝑦‘0)))
6964, 68syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (∀𝑖 ∈ (0..^𝑁)(𝑥𝑖) = (𝑦𝑖) → (𝑥‘0) = (𝑦‘0)))
7061, 69sylbid 229 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → (𝑥‘0) = (𝑦‘0)))
7170imp 444 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥‘0) = (𝑦‘0))
72 simpr 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ( lastS ‘𝑥) = (𝑥‘0))
73 simpr 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ( lastS ‘𝑦) = (𝑦‘0))
7472, 73eqeqan12rd 2628 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (( lastS ‘𝑥) = ( lastS ‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
7574ad2antlr 759 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (( lastS ‘𝑥) = ( lastS ‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
7671, 75mpbird 246 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → ( lastS ‘𝑥) = ( lastS ‘𝑦))
7722, 39, 76jca32 556 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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))))
7841adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → 𝑥 ∈ Word 𝑉)
7978adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑥 ∈ Word 𝑉)
8040adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → 𝑦 ∈ Word 𝑉)
8180adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑦 ∈ Word 𝑉)
82 1red 9934 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ → 1 ∈ ℝ)
83 nngt0 10926 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ → 0 < 𝑁)
84 0lt1 10429 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 < 1
8584a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ → 0 < 1)
8648, 82, 83, 85addgt0d 10481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
87 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑥) = (𝑁 + 1) → (0 < (#‘𝑥) ↔ 0 < (𝑁 + 1)))
8886, 87syl5ibr 235 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
8988ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
9089adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
9190impcom 445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 0 < (#‘𝑥))
9279, 81, 913jca 1235 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉 ∧ 0 < (#‘𝑥)))
9392adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉 ∧ 0 < (#‘𝑥)))
94 2swrd1eqwrdeq 13306 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉 ∧ 0 < (#‘𝑥)) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ ( lastS ‘𝑥) = ( lastS ‘𝑦)))))
9593, 94syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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)))))
9677, 95mpbird 246 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → 𝑥 = 𝑦)
9796exp31 628 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
98973ad2ant3 1077 . . . . . . . . . . . . . . . 16 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
9998expdcom 454 . . . . . . . . . . . . . . 15 (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))))
10099ex 449 . . . . . . . . . . . . . 14 ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
1011003adant3 1074 . . . . . . . . . . . . 13 ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦𝑖), (𝑦‘(𝑖 + 1))} ∈ ran 𝐸) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
10218, 101syl 17 . . . . . . . . . . . 12 (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
103102imp 444 . . . . . . . . . . 11 ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))))
104103expdcom 454 . . . . . . . . . 10 ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
1051043adant3 1074 . . . . . . . . 9 ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥𝑖), (𝑥‘(𝑖 + 1))} ∈ ran 𝐸) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
10617, 105syl 17 . . . . . . . 8 (𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
107106imp31 447 . . . . . . 7 (((𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0))) → ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
108107com12 32 . . . . . 6 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → (((𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
10916, 108syl5bi 231 . . . . 5 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ((𝑥𝐷𝑦𝐷) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
110109imp 444 . . . 4 (((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ (𝑥𝐷𝑦𝐷)) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))
1117, 110sylbid 229 . . 3 (((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ (𝑥𝐷𝑦𝐷)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
112111ralrimivva 2954 . 2 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
113 dff13 6416 . 2 (𝐹:𝐷1-1→((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∀𝑥𝐷𝑦𝐷 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1143, 112, 113sylanbrc 695 1 ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))
 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  ran crn 5039  ⟶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   WWalksN cwwlkn 26206   ClWWalksN cclwwlkn 26277 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-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  clwwlkf1o  26326
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