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Theorem clwlksf1clwwlklemOLD 27242
 Description: Obsolete version of clwlkclwwlkf1lem3 27149 as of 24-May-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
clwlksfclwwlk.1 𝐴 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
Assertion
Ref Expression
clwlksf1clwwlklemOLD ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩) → ∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝑊,𝑐   𝐶,𝑐   𝐹,𝑐   𝑦,𝐺   𝑦,𝑁   𝑈,𝑐,𝑦   𝑦,𝑊
Allowed substitution hints:   𝐴(𝑦,𝑐)   𝐵(𝑦,𝑐)   𝐶(𝑦)   𝐹(𝑦)

Proof of Theorem clwlksf1clwwlklemOLD
StepHypRef Expression
1 clwlksfclwwlk.1 . . . . . . . . . . . 12 𝐴 = (1st𝑐)
2 clwlksfclwwlk.2 . . . . . . . . . . . 12 𝐵 = (2nd𝑐)
3 clwlksfclwwlk.c . . . . . . . . . . . 12 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4 clwlksfclwwlk.f . . . . . . . . . . . 12 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
51, 2, 3, 4clwlksf1clwwlklem3OLD 27241 . . . . . . . . . . 11 (𝑊𝐶 → (2nd𝑊) ∈ Word (Vtx‘𝐺))
61, 2, 3, 4clwlksf1clwwlklem3OLD 27241 . . . . . . . . . . 11 (𝑈𝐶 → (2nd𝑈) ∈ Word (Vtx‘𝐺))
75, 6anim12ci 601 . . . . . . . . . 10 ((𝑊𝐶𝑈𝐶) → ((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)))
87adantr 466 . . . . . . . . 9 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → ((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)))
9 nnnn0 11499 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
109adantl 467 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
111, 2, 3, 4clwlksf1clwwlklem1OLD 27239 . . . . . . . . . . . 12 (𝑈𝐶𝑁 ≤ (♯‘(2nd𝑈)))
1211adantl 467 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (♯‘(2nd𝑈)))
1312adantr 466 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (♯‘(2nd𝑈)))
141, 2, 3, 4clwlksf1clwwlklem1OLD 27239 . . . . . . . . . . . 12 (𝑊𝐶𝑁 ≤ (♯‘(2nd𝑊)))
1514adantr 466 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (♯‘(2nd𝑊)))
1615adantr 466 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (♯‘(2nd𝑊)))
1710, 13, 163jca 1122 . . . . . . . . 9 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊))))
188, 17jca 501 . . . . . . . 8 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))))
1918exp31 406 . . . . . . 7 (𝑊𝐶 → (𝑈𝐶 → (𝑁 ∈ ℕ → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))))))
20193imp31 1103 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))))
2120adantr 466 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))))
221, 2, 3, 4clwlksfclwwlk1hashnOLD 27233 . . . . . . . . . 10 (𝑈𝐶 → (♯‘(1st𝑈)) = 𝑁)
23223ad2ant2 1128 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (♯‘(1st𝑈)) = 𝑁)
2423opeq2d 4546 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (♯‘(1st𝑈))⟩ = ⟨0, 𝑁⟩)
2524oveq2d 6807 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑈) substr ⟨0, 𝑁⟩))
261, 2, 3, 4clwlksfclwwlk1hashnOLD 27233 . . . . . . . . . 10 (𝑊𝐶 → (♯‘(1st𝑊)) = 𝑁)
27263ad2ant3 1129 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (♯‘(1st𝑊)) = 𝑁)
2827opeq2d 4546 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (♯‘(1st𝑊))⟩ = ⟨0, 𝑁⟩)
2928oveq2d 6807 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
3025, 29eqeq12d 2786 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩) ↔ ((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩)))
3130biimpa 462 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → ((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
32 simpl 468 . . . . . . 7 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))) → ((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)))
33 id 22 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
3433, 33jca 501 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
35343ad2ant1 1127 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊))) → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
3635adantl 467 . . . . . . 7 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))) → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
37 3simpc 1146 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊))) → (𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊))))
3837adantl 467 . . . . . . 7 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))) → (𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊))))
39 swrdeq 13646 . . . . . . 7 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))) → (((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩) ↔ (𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))))
4032, 36, 38, 39syl3anc 1476 . . . . . 6 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))) → (((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩) ↔ (𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))))
41 simpr 471 . . . . . 6 ((𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))
4240, 41syl6bi 243 . . . . 5 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (♯‘(2nd𝑈)) ∧ 𝑁 ≤ (♯‘(2nd𝑊)))) → (((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩) → ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))
4321, 31, 42sylc 65 . . . 4 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))
44 lbfzo0 12709 . . . . . . . . 9 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
4544biimpri 218 . . . . . . . 8 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
46453ad2ant1 1127 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → 0 ∈ (0..^𝑁))
4746adantr 466 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → 0 ∈ (0..^𝑁))
48 fveq2 6330 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘0))
49 fveq2 6330 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘0))
5048, 49eqeq12d 2786 . . . . . . 7 (𝑦 = 0 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
5150rspcv 3456 . . . . . 6 (0 ∈ (0..^𝑁) → (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) → ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
5247, 51syl 17 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) → ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
531, 2, 3, 4clwlksf1clwwlklem2OLD 27240 . . . . . . . 8 (𝑈𝐶 → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
54533ad2ant2 1128 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
5554adantr 466 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
561, 2, 3, 4clwlksf1clwwlklem2OLD 27240 . . . . . . . 8 (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
57563ad2ant3 1129 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
5857adantr 466 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
5955, 58eqeq12d 2786 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → (((2nd𝑈)‘0) = ((2nd𝑊)‘0) ↔ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
6052, 59sylibd 229 . . . 4 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) → ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
6143, 60jcai 506 . . 3 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ∧ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
62 elnn0uz 11925 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
639, 62sylib 208 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘0))
64633ad2ant1 1127 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → 𝑁 ∈ (ℤ‘0))
6564adantr 466 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → 𝑁 ∈ (ℤ‘0))
66 fzisfzounsn 12781 . . . . . 6 (𝑁 ∈ (ℤ‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
6765, 66syl 17 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
6867raleqdv 3293 . . . 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)))
69 simpl1 1227 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → 𝑁 ∈ ℕ)
70 fveq2 6330 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘𝑁))
71 fveq2 6330 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘𝑁))
7270, 71eqeq12d 2786 . . . . . 6 (𝑦 = 𝑁 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
7372ralunsn 4560 . . . . 5 (𝑁 ∈ ℕ → (∀𝑦 ∈ ((0..^𝑁) ∪ {𝑁})((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ∧ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁))))
7469, 73syl 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))))
7568, 74bitrd 268 . . 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))))
7661, 75mpbird 247 . 2 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (♯‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (♯‘(1st𝑊))⟩)) → ∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))
7776ex 397 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 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  {crab 3065   ∪ cun 3721  {csn 4316  ⟨cop 4322   class class class wbr 4786   ↦ cmpt 4863  ‘cfv 6029  (class class class)co 6791  1st c1st 7311  2nd c2nd 7312  0cc0 10136   ≤ cle 10275  ℕcn 11220  ℕ0cn0 11492  ℤ≥cuz 11886  ...cfz 12526  ..^cfzo 12666  ♯chash 13314  Word cword 13480   substr csubstr 13484  Vtxcvtx 26088  ClWalkscclwlks 26894 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-map 8009  df-pm 8010  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-card 8963  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-n0 11493  df-z 11578  df-uz 11887  df-fz 12527  df-fzo 12667  df-hash 13315  df-word 13488  df-substr 13492  df-wlks 26723  df-clwlks 26895 This theorem is referenced by:  clwlksf1clwwlkOLD  27243
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