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Theorem clwlksf1clwwlklem 26834
 Description: Lemma for clwlksf1clwwlk 26835. (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 26833 . . . . . . . . . . 11 (𝑊𝐶 → (2nd𝑊) ∈ Word (Vtx‘𝐺))
61, 2, 3, 4clwlksf1clwwlklem3 26833 . . . . . . . . . . 11 (𝑈𝐶 → (2nd𝑈) ∈ Word (Vtx‘𝐺))
75, 6anim12ci 590 . . . . . . . . . 10 ((𝑊𝐶𝑈𝐶) → ((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)))
87adantr 481 . . . . . . . . 9 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → ((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)))
9 nnnn0 11243 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
109adantl 482 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
111, 2, 3, 4clwlksf1clwwlklem1 26831 . . . . . . . . . . . 12 (𝑈𝐶𝑁 ≤ (#‘(2nd𝑈)))
1211adantl 482 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (#‘(2nd𝑈)))
1312adantr 481 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd𝑈)))
141, 2, 3, 4clwlksf1clwwlklem1 26831 . . . . . . . . . . . 12 (𝑊𝐶𝑁 ≤ (#‘(2nd𝑊)))
1514adantr 481 . . . . . . . . . . 11 ((𝑊𝐶𝑈𝐶) → 𝑁 ≤ (#‘(2nd𝑊)))
1615adantr 481 . . . . . . . . . 10 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd𝑊)))
1710, 13, 163jca 1240 . . . . . . . . 9 (((𝑊𝐶𝑈𝐶) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
188, 17jca 554 . . . . . . . 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 1255 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))))
2120adantr 481 . . . . 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 26825 . . . . . . . . . 10 (𝑈𝐶 → (#‘(1st𝑈)) = 𝑁)
23223ad2ant2 1081 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (#‘(1st𝑈)) = 𝑁)
2423opeq2d 4377 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (#‘(1st𝑈))⟩ = ⟨0, 𝑁⟩)
2524oveq2d 6620 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑈) substr ⟨0, 𝑁⟩))
261, 2, 3, 4clwlksfclwwlk1hashn 26825 . . . . . . . . . 10 (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)
27263ad2ant3 1082 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (#‘(1st𝑊)) = 𝑁)
2827opeq2d 4377 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ⟨0, (#‘(1st𝑊))⟩ = ⟨0, 𝑁⟩)
2928oveq2d 6620 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
3025, 29eqeq12d 2636 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) ↔ ((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩)))
3130biimpa 501 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑈) substr ⟨0, 𝑁⟩) = ((2nd𝑊) substr ⟨0, 𝑁⟩))
32 simpl 473 . . . . . . 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 554 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
35343ad2ant1 1080 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))) → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
3635adantl 482 . . . . . . 7 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (𝑁 ∈ ℕ0𝑁 ∈ ℕ0))
37 3simpc 1058 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))) → (𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
3837adantl 482 . . . . . . 7 ((((2nd𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊)))) → (𝑁 ≤ (#‘(2nd𝑈)) ∧ 𝑁 ≤ (#‘(2nd𝑊))))
39 swrdeq 13382 . . . . . . 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 1323 . . . . . 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 477 . . . . . 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 12448 . . . . . . . . 9 (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
4544biimpri 218 . . . . . . . 8 (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
46453ad2ant1 1080 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → 0 ∈ (0..^𝑁))
4746adantr 481 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 0 ∈ (0..^𝑁))
48 fveq2 6148 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘0))
49 fveq2 6148 . . . . . . . 8 (𝑦 = 0 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘0))
5048, 49eqeq12d 2636 . . . . . . 7 (𝑦 = 0 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘0) = ((2nd𝑊)‘0)))
5150rspcv 3291 . . . . . 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 26832 . . . . . . . 8 (𝑈𝐶 → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
54533ad2ant2 1081 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
5554adantr 481 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑈)‘0) = ((2nd𝑈)‘𝑁))
561, 2, 3, 4clwlksf1clwwlklem2 26832 . . . . . . . 8 (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
57563ad2ant3 1082 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
5857adantr 481 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
5955, 58eqeq12d 2636 . . . . 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 11669 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ (ℤ‘0))
639, 62sylib 208 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ‘0))
64633ad2ant1 1080 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → 𝑁 ∈ (ℤ‘0))
6564adantr 481 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 𝑁 ∈ (ℤ‘0))
66 fzisfzounsn 12520 . . . . . 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 3133 . . . 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 1062 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) ∧ ((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩)) → 𝑁 ∈ ℕ)
70 fveq2 6148 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑈)‘𝑦) = ((2nd𝑈)‘𝑁))
71 fveq2 6148 . . . . . . 7 (𝑦 = 𝑁 → ((2nd𝑊)‘𝑦) = ((2nd𝑊)‘𝑁))
7270, 71eqeq12d 2636 . . . . . 6 (𝑦 = 𝑁 → (((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦) ↔ ((2nd𝑈)‘𝑁) = ((2nd𝑊)‘𝑁)))
7372ralunsn 4390 . . . . 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 450 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911   ∪ cun 3553  {csn 4148  ⟨cop 4154   class class class wbr 4613   ↦ cmpt 4673  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  0cc0 9880   ≤ cle 10019  ℕcn 10964  ℕ0cn0 11236  ℤ≥cuz 11631  ...cfz 12268  ..^cfzo 12406  #chash 13057  Word cword 13230   substr csubstr 13234  Vtxcvtx 25774  ClWalkscclwlks 26535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-substr 13242  df-wlks 26365  df-clwlks 26536 This theorem is referenced by:  clwlksf1clwwlk  26835
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