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Theorem clwlkf1clwwlklem 26376
 Description: Lemma for clwlkf1clwwlk 26377. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
Hypotheses
Ref Expression
clwlkfclwwlk.1 𝐴 = (1st𝑐)
clwlkfclwwlk.2 𝐵 = (2nd𝑐)
clwlkfclwwlk.c 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
clwlkfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlkf1clwwlklem ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) → ∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))
Distinct variable groups:   𝐸,𝑐   𝑁,𝑐   𝑉,𝑐   𝑊,𝑐   𝐶,𝑐   𝐹,𝑐   𝑦,𝑁   𝑈,𝑐,𝑦   𝑦,𝑉   𝑦,𝑊
Allowed substitution hints:   𝐴(𝑦,𝑐)   𝐵(𝑦,𝑐)   𝐶(𝑦)   𝐸(𝑦)   𝐹(𝑦)

Proof of Theorem clwlkf1clwwlklem
StepHypRef Expression
1 clwlkfclwwlk.1 . . . . . . . . . . . . 13 𝐴 = (1st𝑐)
2 clwlkfclwwlk.2 . . . . . . . . . . . . 13 𝐵 = (2nd𝑐)
3 clwlkfclwwlk.c . . . . . . . . . . . . 13 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
4 clwlkfclwwlk.f . . . . . . . . . . . . 13 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
51, 2, 3, 4clwlkf1clwwlklem3 26375 . . . . . . . . . . . 12 (𝑊𝐶 → (2nd𝑊) ∈ Word 𝑉)
61, 2, 3, 4clwlkf1clwwlklem3 26375 . . . . . . . . . . . 12 (𝑈𝐶 → (2nd𝑈) ∈ Word 𝑉)
75, 6anim12ci 589 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → ((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉))
87adantr 480 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → ((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉))
9 nnnn0 11176 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
109adantl 481 . . . . . . . . . . 11 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
111, 2, 3, 4clwlkf1clwwlklem1 26373 . . . . . . . . . . . . 13 (𝑈𝐶𝑁 ≤ (#‘(2nd𝑈)))
1211adantl 481 . . . . . . . . . . . 12 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (#‘(2nd𝑈)))
1312adantr 480 . . . . . . . . . . 11 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd𝑈)))
141, 2, 3, 4clwlkf1clwwlklem1 26373 . . . . . . . . . . . . 13 (𝑊𝐶𝑁 ≤ (#‘(2nd𝑊)))
1514adantr 480 . . . . . . . . . . . 12 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (#‘(2nd𝑊)))
1615adantr 480 . . . . . . . . . . 11 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd𝑊)))
1710, 13, 163jca 1235 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
188, 17jca 553 . . . . . . . . 9 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → (((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))
1918exp31 628 . . . . . . . 8 (𝑊𝐶 → (𝑈𝐶 → (𝑁 ∈ ℕ → (((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))))
2019com13 86 . . . . . . 7 (𝑁 ∈ ℕ → (𝑈𝐶 → (𝑊𝐶 → (((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))))
21203imp 1249 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))
2221adantr 480 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))
231, 2, 3, 4clwlkfclwwlk1hashn 26368 . . . . . . . . . 10 (𝑈𝐶 → (#‘(1st𝑈)) = 𝑁)
24233ad2ant2 1076 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (#‘(1st𝑈)) = 𝑁)
2524opeq2d 4347 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (#‘(1st𝑈))⟩ = ⟨0, 𝑁⟩)
2625oveq2d 6565 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑈) substr ⟨0, 𝑁⟩))
271, 2, 3, 4clwlkfclwwlk1hashn 26368 . . . . . . . . . 10 (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)
28273ad2ant3 1077 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (#‘(1st𝑊)) = 𝑁)
2928opeq2d 4347 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (#‘(1st𝑊))⟩ = ⟨0, 𝑁⟩)
3029oveq2d 6565 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
3126, 30eqeq12d 2625 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) ↔ ((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩)))
3231biimpa 500 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
33 simpl 472 . . . . . . 7 ((((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → ((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉))
34 id 22 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
3534, 34jca 553 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
36353ad2ant1 1075 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))) → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
3736adantl 481 . . . . . . 7 ((((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
38 3simpc 1053 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))) → (𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
3938adantl 481 . . . . . . 7 ((((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
40 swrdeq 13296 . . . . . . 7 ((((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩) ↔ (𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))))
4133, 37, 39, 40syl3anc 1318 . . . . . 6 ((((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩) ↔ (𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))))
42 simpr 476 . . . . . 6 ((𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))
4341, 42syl6bi 242 . . . . 5 ((((2nd𝑈) ∈ Word 𝑉 ∧ (2nd𝑊) ∈ Word 𝑉) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩) → ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))
4422, 32, 43sylc 63 . . . 4 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))
45 lbfzo0 12375 . . . . . . . . 9 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
4645biimpri 217 . . . . . . . 8 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
47463ad2ant1 1075 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → 0 ∈ (0..^𝑁))
4847adantr 480 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 0 ∈ (0..^𝑁))
49 fveq2 6103 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘0))
50 fveq2 6103 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘0))
5149, 50eqeq12d 2625 . . . . . . 7 (𝑦 = 0 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
5251rspcv 3278 . . . . . 6 (0 ∈ (0..^𝑁) → (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) → ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
5348, 52syl 17 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) → ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
541, 2, 3, 4clwlkf1clwwlklem2 26374 . . . . . . . 8 (𝑈𝐶 → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
55543ad2ant2 1076 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
5655adantr 480 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
571, 2, 3, 4clwlkf1clwwlklem2 26374 . . . . . . . 8 (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
58573ad2ant3 1077 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
5958adantr 480 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
6056, 59eqeq12d 2625 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (((2nd𝑈)‘0) = ((2nd𝑊)‘0) ↔ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
6153, 60sylibd 228 . . . 4 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) → ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
6244, 61jcai 557 . . 3 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ∧ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
63 elnn0uz 11601 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
649, 63sylib 207 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘0))
65643ad2ant1 1075 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → 𝑁 ∈ (ℤ‘0))
6665adantr 480 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 𝑁 ∈ (ℤ‘0))
67 fzisfzounsn 12445 . . . . . 6 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
6866, 67syl 17 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
6968raleqdv 3121 . . . 4 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ∀𝑦 ∈ ((0..^𝑁) ∪ {𝑁})((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))
70 simpl1 1057 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 𝑁 ∈ ℕ)
71 fveq2 6103 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘𝑁))
72 fveq2 6103 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘𝑁))
7371, 72eqeq12d 2625 . . . . . 6 (𝑦 = 𝑁 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
7473ralunsn 4360 . . . . 5 (𝑁 ∈ ℕ → (∀𝑦 ∈ ((0..^𝑁) ∪ {𝑁})((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ∧ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁))))
7570, 74syl 17 . . . 4 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (∀𝑦 ∈ ((0..^𝑁) ∪ {𝑁})((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ∧ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁))))
7669, 75bitrd 267 . . 3 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ∧ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁))))
7762, 76mpbird 246 . 2 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))
7877ex 449 1 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) → ∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∪ cun 3538  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815   ≤ cle 9954  ℕcn 10897  ℕ0cn0 11169  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   substr csubstr 13150   ClWalks cclwlk 26275 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-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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-substr 13158  df-wlk 26036  df-clwlk 26278 This theorem is referenced by:  clwlkf1clwwlk  26377
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