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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  cn 11220  0cn0 11492  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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