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 = (1st ‘𝑐)
clwlksfclwwlk.2	⊢ 𝐵 = (2nd ‘𝑐)
clwlksfclwwlk.c	⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))

Assertion

Ref	Expression
clwlksf1clwwlklem	⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩) → ∀𝑦 ∈ (0...𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘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

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . . . . . . . . . . 12 ⊢ 𝐴 = (1st ‘𝑐)
2		clwlksfclwwlk.2	. . . . . . . . . . . 12 ⊢ 𝐵 = (2nd ‘𝑐)
3		clwlksfclwwlk.c	. . . . . . . . . . . 12 ⊢ 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
4		clwlksfclwwlk.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
5	1, 2, 3, 4	clwlksf1clwwlklem3 41274	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐶 → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺))
6	1, 2, 3, 4	clwlksf1clwwlklem3 41274	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝐶 → (2nd ‘𝑈) ∈ Word (Vtx‘𝐺))
7	5, 6	anim12ci 589	. . . . . . . . . 10 ⊢ ((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) → ((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)))
8	7	adantr 480	. . . . . . . . 9 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → ((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)))
9		nnnn0 11176	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
10	9	adantl 481	. . . . . . . . . 10 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
11	1, 2, 3, 4	clwlksf1clwwlklem1 41272	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐶 → 𝑁 ≤ (#‘(2nd ‘𝑈)))
12	11	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) → 𝑁 ≤ (#‘(2nd ‘𝑈)))
13	12	adantr 480	. . . . . . . . . 10 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd ‘𝑈)))
14	1, 2, 3, 4	clwlksf1clwwlklem1 41272	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ 𝐶 → 𝑁 ≤ (#‘(2nd ‘𝑊)))
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) → 𝑁 ≤ (#‘(2nd ‘𝑊)))
16	15	adantr 480	. . . . . . . . . 10 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd ‘𝑊)))
17	10, 13, 16	3jca 1235	. . . . . . . . 9 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))))
18	8, 17	jca 553	. . . . . . . 8 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → (((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))))
19	18	exp31 628	. . . . . . 7 ⊢ (𝑊 ∈ 𝐶 → (𝑈 ∈ 𝐶 → (𝑁 ∈ ℕ → (((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))))))
20	19	3imp31 1250	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))))
21	20	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))))
22	1, 2, 3, 4	clwlksfclwwlk1hashn 41266	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐶 → (#‘(1st ‘𝑈)) = 𝑁)
23	22	3ad2ant2 1076	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (#‘(1st ‘𝑈)) = 𝑁)
24	23	opeq2d 4347	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ⟨0, (#‘(1st ‘𝑈))⟩ = ⟨0, 𝑁⟩)
25	24	oveq2d 6565	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑈) substr ⟨0, 𝑁⟩))
26	1, 2, 3, 4	clwlksfclwwlk1hashn 41266	. . . . . . . . . 10 ⊢ (𝑊 ∈ 𝐶 → (#‘(1st ‘𝑊)) = 𝑁)
27	26	3ad2ant3 1077	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (#‘(1st ‘𝑊)) = 𝑁)
28	27	opeq2d 4347	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ⟨0, (#‘(1st ‘𝑊))⟩ = ⟨0, 𝑁⟩)
29	28	oveq2d 6565	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩) = ((2nd ‘𝑊) substr ⟨0, 𝑁⟩))
30	25, 29	eqeq12d 2625	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩) ↔ ((2nd ‘𝑈) substr ⟨0, 𝑁⟩) = ((2nd ‘𝑊) substr ⟨0, 𝑁⟩)))
31	30	biimpa 500	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → ((2nd ‘𝑈) substr ⟨0, 𝑁⟩) = ((2nd ‘𝑊) substr ⟨0, 𝑁⟩))
32		simpl 472	. . . . . . 7 ⊢ ((((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))) → ((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)))
33		id 22	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0)
34	33, 33	jca 553	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0))
35	34	3ad2ant1 1075	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))) → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0))
36	35	adantl 481	. . . . . . 7 ⊢ ((((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))) → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0))
37		3simpc 1053	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))) → (𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))))
38	37	adantl 481	. . . . . . 7 ⊢ ((((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))) → (𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))))
39		swrdeq 13296	. . . . . . 7 ⊢ ((((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))) → (((2nd ‘𝑈) substr ⟨0, 𝑁⟩) = ((2nd ‘𝑊) substr ⟨0, 𝑁⟩) ↔ (𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦))))
40	32, 36, 38, 39	syl3anc 1318	. . . . . 6 ⊢ ((((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))) → (((2nd ‘𝑈) substr ⟨0, 𝑁⟩) = ((2nd ‘𝑊) substr ⟨0, 𝑁⟩) ↔ (𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦))))
41		simpr 476	. . . . . 6 ⊢ ((𝑁 = 𝑁 ∧ ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦))
42	40, 41	syl6bi 242	. . . . 5 ⊢ ((((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊)))) → (((2nd ‘𝑈) substr ⟨0, 𝑁⟩) = ((2nd ‘𝑊) substr ⟨0, 𝑁⟩) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦)))
43	21, 31, 42	sylc 63	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦))
44		lbfzo0 12375	. . . . . . . . 9 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
45	44	biimpri 217	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
46	45	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → 0 ∈ (0..^𝑁))
47	46	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → 0 ∈ (0..^𝑁))
48		fveq2 6103	. . . . . . . 8 ⊢ (𝑦 = 0 → ((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑈)‘0))
49		fveq2 6103	. . . . . . . 8 ⊢ (𝑦 = 0 → ((2nd ‘𝑊)‘𝑦) = ((2nd ‘𝑊)‘0))
50	48, 49	eqeq12d 2625	. . . . . . 7 ⊢ (𝑦 = 0 → (((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ↔ ((2nd ‘𝑈)‘0) = ((2nd ‘𝑊)‘0)))
51	50	rspcv 3278	. . . . . 6 ⊢ (0 ∈ (0..^𝑁) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) → ((2nd ‘𝑈)‘0) = ((2nd ‘𝑊)‘0)))
52	47, 51	syl 17	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) → ((2nd ‘𝑈)‘0) = ((2nd ‘𝑊)‘0)))
53	1, 2, 3, 4	clwlksf1clwwlklem2 41273	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐶 → ((2nd ‘𝑈)‘0) = ((2nd ‘𝑈)‘𝑁))
54	53	3ad2ant2 1076	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((2nd ‘𝑈)‘0) = ((2nd ‘𝑈)‘𝑁))
55	54	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → ((2nd ‘𝑈)‘0) = ((2nd ‘𝑈)‘𝑁))
56	1, 2, 3, 4	clwlksf1clwwlklem2 41273	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐶 → ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘𝑁))
57	56	3ad2ant3 1077	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘𝑁))
58	57	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘𝑁))
59	55, 58	eqeq12d 2625	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (((2nd ‘𝑈)‘0) = ((2nd ‘𝑊)‘0) ↔ ((2nd ‘𝑈)‘𝑁) = ((2nd ‘𝑊)‘𝑁)))
60	52, 59	sylibd 228	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) → ((2nd ‘𝑈)‘𝑁) = ((2nd ‘𝑊)‘𝑁)))
61	43, 60	jcai 557	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ∧ ((2nd ‘𝑈)‘𝑁) = ((2nd ‘𝑊)‘𝑁)))
62		elnn0uz 11601	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
63	9, 62	sylib 207	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘0))
64	63	3ad2ant1 1075	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → 𝑁 ∈ (ℤ≥‘0))
65	64	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → 𝑁 ∈ (ℤ≥‘0))
66		fzisfzounsn 12445	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
67	65, 66	syl 17	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
68	67	raleqdv 3121	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ↔ ∀𝑦 ∈ ((0..^𝑁) ∪ {𝑁})((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦)))
69		simpl1 1057	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → 𝑁 ∈ ℕ)
70		fveq2 6103	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑈)‘𝑁))
71		fveq2 6103	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((2nd ‘𝑊)‘𝑦) = ((2nd ‘𝑊)‘𝑁))
72	70, 71	eqeq12d 2625	. . . . . 6 ⊢ (𝑦 = 𝑁 → (((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ↔ ((2nd ‘𝑈)‘𝑁) = ((2nd ‘𝑊)‘𝑁)))
73	72	ralunsn 4360	. . . . 5 ⊢ (𝑁 ∈ ℕ → (∀𝑦 ∈ ((0..^𝑁) ∪ {𝑁})((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ∧ ((2nd ‘𝑈)‘𝑁) = ((2nd ‘𝑊)‘𝑁))))
74	69, 73	syl 17	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (∀𝑦 ∈ ((0..^𝑁) ∪ {𝑁})((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ∧ ((2nd ‘𝑈)‘𝑁) = ((2nd ‘𝑊)‘𝑁))))
75	68, 74	bitrd 267	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ∧ ((2nd ‘𝑈)‘𝑁) = ((2nd ‘𝑊)‘𝑁))))
76	61, 75	mpbird 246	. 2 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → ∀𝑦 ∈ (0...𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦))
77	76	ex 449	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩) → ∀𝑦 ∈ (0...𝑁)((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦)))

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  1^st c1st 7057  2^nd 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