Theorem clwlksf1clwwlklem 26968

Description: Lemma for clwlksf1clwwlk 26969. (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 26967	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐶 → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺))
6	1, 2, 3, 4	clwlksf1clwwlklem3 26967	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝐶 → (2nd ‘𝑈) ∈ Word (Vtx‘𝐺))
7	5, 6	anim12ci 591	. . . . . . . . . 10 ⊢ ((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) → ((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)))
8	7	adantr 481	. . . . . . . . 9 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → ((2nd ‘𝑈) ∈ Word (Vtx‘𝐺) ∧ (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)))
9		nnnn0 11299	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
10	9	adantl 482	. . . . . . . . . 10 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
11	1, 2, 3, 4	clwlksf1clwwlklem1 26965	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐶 → 𝑁 ≤ (#‘(2nd ‘𝑈)))
12	11	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) → 𝑁 ≤ (#‘(2nd ‘𝑈)))
13	12	adantr 481	. . . . . . . . . 10 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd ‘𝑈)))
14	1, 2, 3, 4	clwlksf1clwwlklem1 26965	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ 𝐶 → 𝑁 ≤ (#‘(2nd ‘𝑊)))
15	14	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) → 𝑁 ≤ (#‘(2nd ‘𝑊)))
16	15	adantr 481	. . . . . . . . . 10 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → 𝑁 ≤ (#‘(2nd ‘𝑊)))
17	10, 13, 16	3jca 1242	. . . . . . . . 9 ⊢ (((𝑊 ∈ 𝐶 ∧ 𝑈 ∈ 𝐶) ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))))
18	8, 17	jca 554	. . . . . . . 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 630	. . . . . . 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 1257	. . . . . 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 481	. . . . 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 26959	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐶 → (#‘(1st ‘𝑈)) = 𝑁)
23	22	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (#‘(1st ‘𝑈)) = 𝑁)
24	23	opeq2d 4409	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ⟨0, (#‘(1st ‘𝑈))⟩ = ⟨0, 𝑁⟩)
25	24	oveq2d 6666	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑈) substr ⟨0, 𝑁⟩))
26	1, 2, 3, 4	clwlksfclwwlk1hashn 26959	. . . . . . . . . 10 ⊢ (𝑊 ∈ 𝐶 → (#‘(1st ‘𝑊)) = 𝑁)
27	26	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (#‘(1st ‘𝑊)) = 𝑁)
28	27	opeq2d 4409	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ⟨0, (#‘(1st ‘𝑊))⟩ = ⟨0, 𝑁⟩)
29	28	oveq2d 6666	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩) = ((2nd ‘𝑊) substr ⟨0, 𝑁⟩))
30	25, 29	eqeq12d 2637	. . . . . 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 501	. . . . 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 473	. . . . . . 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 554	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0))
35	34	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))) → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0))
36	35	adantl 482	. . . . . . 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 1060	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))) → (𝑁 ≤ (#‘(2nd ‘𝑈)) ∧ 𝑁 ≤ (#‘(2nd ‘𝑊))))
38	37	adantl 482	. . . . . . 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 13444	. . . . . . 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 1326	. . . . . 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 477	. . . . . 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 243	. . . . 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 65	. . . 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 12507	. . . . . . . . 9 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
45	44	biimpri 218	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
46	45	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → 0 ∈ (0..^𝑁))
47	46	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → 0 ∈ (0..^𝑁))
48		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 0 → ((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑈)‘0))
49		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 0 → ((2nd ‘𝑊)‘𝑦) = ((2nd ‘𝑊)‘0))
50	48, 49	eqeq12d 2637	. . . . . . 7 ⊢ (𝑦 = 0 → (((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ↔ ((2nd ‘𝑈)‘0) = ((2nd ‘𝑊)‘0)))
51	50	rspcv 3305	. . . . . 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 26966	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐶 → ((2nd ‘𝑈)‘0) = ((2nd ‘𝑈)‘𝑁))
54	53	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((2nd ‘𝑈)‘0) = ((2nd ‘𝑈)‘𝑁))
55	54	adantr 481	. . . . . 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 26966	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐶 → ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘𝑁))
57	56	3ad2ant3 1084	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘𝑁))
58	57	adantr 481	. . . . . 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 2637	. . . . 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 229	. . . 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 559	. . 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 11725	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
63	9, 62	sylib 208	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘0))
64	63	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → 𝑁 ∈ (ℤ≥‘0))
65	64	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → 𝑁 ∈ (ℤ≥‘0))
66		fzisfzounsn 12580	. . . . . 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 3144	. . . 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 1064	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) ∧ ((2nd ‘𝑈) substr ⟨0, (#‘(1st ‘𝑈))⟩) = ((2nd ‘𝑊) substr ⟨0, (#‘(1st ‘𝑊))⟩)) → 𝑁 ∈ ℕ)
70		fveq2 6191	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑈)‘𝑁))
71		fveq2 6191	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((2nd ‘𝑊)‘𝑦) = ((2nd ‘𝑊)‘𝑁))
72	70, 71	eqeq12d 2637	. . . . . 6 ⊢ (𝑦 = 𝑁 → (((2nd ‘𝑈)‘𝑦) = ((2nd ‘𝑊)‘𝑦) ↔ ((2nd ‘𝑈)‘𝑁) = ((2nd ‘𝑊)‘𝑁)))
73	72	ralunsn 4422	. . . . 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 268	. . 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 247	. 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 450	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ∪ cun 3572  {csn 4177  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  0cc0 9936   ≤ cle 10075  ℕcn 11020  ℕ₀cn0 11292  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465  #chash 13117  Word cword 13291   substr csubstr 13295  Vtxcvtx 25874  ClWalkscclwlks 26666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-substr 13303  df-wlks 26495  df-clwlks 26667

This theorem is referenced by:  clwlksf1clwwlk  26969