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Theorem clwlksf1clwwlklem 41275
 Description: Lemma for clwlksf1clwwlk 41276. (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 41274 . . . . . . . . . . 11 (𝑊𝐶 → (2nd𝑊) ∈ Word (Vtx‘𝐺))
61, 2, 3, 4clwlksf1clwwlklem3 41274 . . . . . . . . . . 11 (𝑈𝐶 → (2nd𝑈) ∈ Word (Vtx‘𝐺))
75, 6anim12ci 589 . . . . . . . . . 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 11176 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
109adantl 481 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
111, 2, 3, 4clwlksf1clwwlklem1 41272 . . . . . . . . . . . 12 (𝑈𝐶𝑁 ≤ (#‘(2nd𝑈)))
1211adantl 481 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (#‘(2nd𝑈)))
1312adantr 480 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd𝑈)))
141, 2, 3, 4clwlksf1clwwlklem1 41272 . . . . . . . . . . . 12 (𝑊𝐶𝑁 ≤ (#‘(2nd𝑊)))
1514adantr 480 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (#‘(2nd𝑊)))
1615adantr 480 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd𝑊)))
1710, 13, 163jca 1235 . . . . . . . . 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 628 . . . . . . 7 (𝑊𝐶 → (𝑈𝐶 → (𝑁 ∈ ℕ → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))))
20193imp31 1250 . . . . . 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 41266 . . . . . . . . . 10 (𝑈𝐶 → (#‘(1st𝑈)) = 𝑁)
23223ad2ant2 1076 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (#‘(1st𝑈)) = 𝑁)
2423opeq2d 4347 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (#‘(1st𝑈))⟩ = ⟨0, 𝑁⟩)
2524oveq2d 6565 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑈) substr ⟨0, 𝑁⟩))
261, 2, 3, 4clwlksfclwwlk1hashn 41266 . . . . . . . . . 10 (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)
27263ad2ant3 1077 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (#‘(1st𝑊)) = 𝑁)
2827opeq2d 4347 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (#‘(1st𝑊))⟩ = ⟨0, 𝑁⟩)
2928oveq2d 6565 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
3025, 29eqeq12d 2625 . . . . . 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 1075 . . . . . . . 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 1053 . . . . . . . 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 13296 . . . . . . 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 1318 . . . . . 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 242 . . . . 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 63 . . . 4 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ∀𝑦 ∈ (0..^𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦))
44 lbfzo0 12375 . . . . . . . . 9 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
4544biimpri 217 . . . . . . . 8 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
46453ad2ant1 1075 . . . . . . 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 6103 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘0))
49 fveq2 6103 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘0))
5048, 49eqeq12d 2625 . . . . . . 7 (𝑦 = 0 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
5150rspcv 3278 . . . . . 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 41273 . . . . . . . 8 (𝑈𝐶 → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
54533ad2ant2 1076 . . . . . . 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 41273 . . . . . . . 8 (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
57563ad2ant3 1077 . . . . . . 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 2625 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → (((2nd𝑈)‘0) = ((2nd𝑊)‘0) ↔ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
6052, 59sylibd 228 . . . 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 557 . . 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 11601 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
639, 62sylib 207 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘0))
64633ad2ant1 1075 . . . . . . 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 12445 . . . . . 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 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)))
69 simpl1 1057 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 𝑁 ∈ ℕ)
70 fveq2 6103 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘𝑁))
71 fveq2 6103 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘𝑁))
7270, 71eqeq12d 2625 . . . . . 6 (𝑦 = 𝑁 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
7372ralunsn 4360 . . . . 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 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))))
7661, 75mpbird 246 . 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 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  Vtxcvtx 25673  ClWalkScclwlks 40976 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-ifp 1007  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-1wlks 40800  df-clwlks 40977 This theorem is referenced by:  clwlksf1clwwlk  41276
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