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Theorem clwlksf1clwwlklem 27055
 Description: Lemma for clwlksf1clwwlk 27056. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)
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
clwlksf1clwwlklem ((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 clwlksf1clwwlklem
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, 4clwlksf1clwwlklem3 27054 . . . . . . . . . . 11 (𝑊𝐶 → (2nd𝑊) ∈ Word (Vtx‘𝐺))
61, 2, 3, 4clwlksf1clwwlklem3 27054 . . . . . . . . . . 11 (𝑈𝐶 → (2nd𝑈) ∈ Word (Vtx‘𝐺))
75, 6anim12ci 590 . . . . . . . . . 10 ((𝑊𝐶𝑈𝐶) → ((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)))
87adantr 480 . . . . . . . . 9 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → ((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)))
9 nnnn0 11337 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
109adantl 481 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
111, 2, 3, 4clwlksf1clwwlklem1 27052 . . . . . . . . . . . 12 (𝑈𝐶𝑁 ≤ (#‘(2nd𝑈)))
1211adantl 481 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (#‘(2nd𝑈)))
1312adantr 480 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd𝑈)))
141, 2, 3, 4clwlksf1clwwlklem1 27052 . . . . . . . . . . . 12 (𝑊𝐶𝑁 ≤ (#‘(2nd𝑊)))
1514adantr 480 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (#‘(2nd𝑊)))
1615adantr 480 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd𝑊)))
1710, 13, 163jca 1261 . . . . . . . . 9 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
188, 17jca 553 . . . . . . . 8 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))
1918exp31 629 . . . . . . 7 (𝑊𝐶 → (𝑈𝐶 → (𝑁 ∈ ℕ → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))))
20193imp31 1276 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))
2120adantr 480 . . . . 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, 4clwlksfclwwlk1hashn 27046 . . . . . . . . . 10 (𝑈𝐶 → (#‘(1st𝑈)) = 𝑁)
23223ad2ant2 1103 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (#‘(1st𝑈)) = 𝑁)
2423opeq2d 4440 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (#‘(1st𝑈))⟩ = ⟨0, 𝑁⟩)
2524oveq2d 6706 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑈) substr ⟨0, 𝑁⟩))
261, 2, 3, 4clwlksfclwwlk1hashn 27046 . . . . . . . . . 10 (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)
27263ad2ant3 1104 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (#‘(1st𝑊)) = 𝑁)
2827opeq2d 4440 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (#‘(1st𝑊))⟩ = ⟨0, 𝑁⟩)
2928oveq2d 6706 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
3025, 29eqeq12d 2666 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) ↔ ((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩)))
3130biimpa 500 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
32 simpl 472 . . . . . . 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 553 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
35343ad2ant1 1102 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))) → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
3635adantl 481 . . . . . . 7 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
37 3simpc 1080 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))) → (𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
3837adantl 481 . . . . . . 7 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
39 swrdeq 13490 . . . . . . 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 1366 . . . . . 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 476 . . . . . 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 12547 . . . . . . . . 9 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
4544biimpri 218 . . . . . . . 8 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
46453ad2ant1 1102 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → 0 ∈ (0..^𝑁))
4746adantr 480 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 0 ∈ (0..^𝑁))
48 fveq2 6229 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘0))
49 fveq2 6229 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘0))
5048, 49eqeq12d 2666 . . . . . . 7 (𝑦 = 0 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
5150rspcv 3336 . . . . . 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, 4clwlksf1clwwlklem2 27053 . . . . . . . 8 (𝑈𝐶 → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
54533ad2ant2 1103 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
5554adantr 480 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
561, 2, 3, 4clwlksf1clwwlklem2 27053 . . . . . . . 8 (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
57563ad2ant3 1104 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
5857adantr 480 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
5955, 58eqeq12d 2666 . . . . 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 558 . . 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 11763 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
639, 62sylib 208 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘0))
64633ad2ant1 1102 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → 𝑁 ∈ (ℤ‘0))
6564adantr 480 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 𝑁 ∈ (ℤ‘0))
66 fzisfzounsn 12620 . . . . . 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 3174 . . . 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 1084 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 𝑁 ∈ ℕ)
70 fveq2 6229 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘𝑁))
71 fveq2 6229 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘𝑁))
7270, 71eqeq12d 2666 . . . . . 6 (𝑦 = 𝑁 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
7372ralunsn 4454 . . . . 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 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {crab 2945   ∪ cun 3605  {csn 4210  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  0cc0 9974   ≤ cle 10113  ℕcn 11058  ℕ0cn0 11330  ℤ≥cuz 11725  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323   substr csubstr 13327  Vtxcvtx 25919  ClWalkscclwlks 26722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-substr 13335  df-wlks 26551  df-clwlks 26723 This theorem is referenced by:  clwlksf1clwwlk  27056
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