Theorem clwlkclwwlkf1lem3 30038

Description: Lemma 3 for clwlkclwwlkf1 30042. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 30-Oct-2022.)

Hypotheses

Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
clwlkclwwlkf.a	⊢ 𝐴 = (1st ‘𝑈)
clwlkclwwlkf.b	⊢ 𝐵 = (2nd ‘𝑈)
clwlkclwwlkf.d	⊢ 𝐷 = (1st ‘𝑊)
clwlkclwwlkf.e	⊢ 𝐸 = (2nd ‘𝑊)

Assertion

Ref	Expression
clwlkclwwlkf1lem3	⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))

Distinct variable groups:   𝑖,𝐺   𝑤,𝐺   𝑤,𝐴   𝑤,𝑈   𝐴,𝑖   𝐵,𝑖   𝐷,𝑖   𝑤,𝐷   𝑖,𝐸   𝑤,𝑊

Allowed substitution hints:   𝐵(𝑤)   𝐶(𝑤,𝑖)   𝑈(𝑖)   𝐸(𝑤)   𝑊(𝑖)

Proof of Theorem clwlkclwwlkf1lem3

Step	Hyp	Ref	Expression
1		clwlkclwwlkf.c	. . . . 5 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
2		clwlkclwwlkf.a	. . . . 5 ⊢ 𝐴 = (1st ‘𝑈)
3		clwlkclwwlkf.b	. . . . 5 ⊢ 𝐵 = (2nd ‘𝑈)
4		clwlkclwwlkf.d	. . . . 5 ⊢ 𝐷 = (1st ‘𝑊)
5		clwlkclwwlkf.e	. . . . 5 ⊢ 𝐸 = (2nd ‘𝑊)
6	1, 2, 3, 4, 5	clwlkclwwlkf1lem2 30037	. . . 4 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))
7		simprr 772	. . . . 5 ⊢ (((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))) → ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))
8	1, 2, 3	clwlkclwwlkflem 30036	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
9	1, 4, 5	clwlkclwwlkflem 30036	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
10		lbfzo0 13756	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^(♯‘𝐴)) ↔ (♯‘𝐴) ∈ ℕ)
11	10	biimpri 228	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) ∈ ℕ → 0 ∈ (0..^(♯‘𝐴)))
12	11	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 0 ∈ (0..^(♯‘𝐴)))
13	12	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → 0 ∈ (0..^(♯‘𝐴)))
14	13	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → 0 ∈ (0..^(♯‘𝐴)))
15		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝐵‘𝑖) = (𝐵‘0))
16		fveq2 6920	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝐸‘𝑖) = (𝐸‘0))
17	15, 16	eqeq12d 2756	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → ((𝐵‘𝑖) = (𝐸‘𝑖) ↔ (𝐵‘0) = (𝐸‘0)))
18	17	rspcv 3631	. . . . . . . . . . . 12 ⊢ (0 ∈ (0..^(♯‘𝐴)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) → (𝐵‘0) = (𝐸‘0)))
19	14, 18	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) → (𝐵‘0) = (𝐸‘0)))
20		simpl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐵‘0))
21		eqtr 2763	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
22	21	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
23	20, 22	eqtrd 2780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))
24	23	exp32 420	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵‘(♯‘𝐴)) = (𝐵‘0) → ((𝐵‘0) = (𝐸‘0) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
25	24	com23 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵‘(♯‘𝐴)) = (𝐵‘0) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
26	25	eqcoms 2748	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
27	26	3ad2ant2 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
28	27	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
29	28	3ad2ant2 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
30	29	impcom 407	. . . . . . . . . . . . . . 15 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷))))
31	30	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷))))
32	31	imp 406	. . . . . . . . . . . . 13 ⊢ (((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) ∧ (𝐵‘0) = (𝐸‘0)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))
33		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐷) = (♯‘𝐴) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
34	33	eqcoms 2748	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) = (♯‘𝐷) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
35	34	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
36	35	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) ∧ (𝐵‘0) = (𝐸‘0)) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
37	32, 36	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) ∧ (𝐵‘0) = (𝐸‘0)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))
38	37	ex 412	. . . . . . . . . . 11 ⊢ ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
39	19, 38	syld 47	. . . . . . . . . 10 ⊢ ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
40	39	ex 412	. . . . . . . . 9 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → ((♯‘𝐴) = (♯‘𝐷) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
41	8, 9, 40	syl2an 595	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((♯‘𝐴) = (♯‘𝐷) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
42	41	impd 410	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
43	42	3adant3 1132	. . . . . 6 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
44	43	imp 406	. . . . 5 ⊢ (((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))
45	7, 44	jca 511	. . . 4 ⊢ (((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
46	6, 45	mpdan 686	. . 3 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
47		fvex 6933	. . . 4 ⊢ (♯‘𝐴) ∈ V
48		fveq2 6920	. . . . . 6 ⊢ (𝑖 = (♯‘𝐴) → (𝐵‘𝑖) = (𝐵‘(♯‘𝐴)))
49		fveq2 6920	. . . . . 6 ⊢ (𝑖 = (♯‘𝐴) → (𝐸‘𝑖) = (𝐸‘(♯‘𝐴)))
50	48, 49	eqeq12d 2756	. . . . 5 ⊢ (𝑖 = (♯‘𝐴) → ((𝐵‘𝑖) = (𝐸‘𝑖) ↔ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
51	50	ralunsn 4918	. . . 4 ⊢ ((♯‘𝐴) ∈ V → (∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵‘𝑖) = (𝐸‘𝑖) ↔ (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
52	47, 51	ax-mp 5	. . 3 ⊢ (∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵‘𝑖) = (𝐸‘𝑖) ↔ (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
53	46, 52	sylibr 234	. 2 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵‘𝑖) = (𝐸‘𝑖))
54		nnnn0 12560	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
55		elnn0uz 12948	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ≥‘0))
56	54, 55	sylib 218	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ (ℤ≥‘0))
57	56	3ad2ant3 1135	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ (ℤ≥‘0))
58	8, 57	syl 17	. . . 4 ⊢ (𝑈 ∈ 𝐶 → (♯‘𝐴) ∈ (ℤ≥‘0))
59	58	3ad2ant1 1133	. . 3 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (♯‘𝐴) ∈ (ℤ≥‘0))
60		fzisfzounsn 13829	. . 3 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘0) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
61	59, 60	syl 17	. 2 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
62	53, 61	raleqtrrdv 3338	1 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ∪ cun 3974  {csn 4648   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  0cc0 11184  1c1 11185   ≤ cle 11325  ℕcn 12293  ℕ₀cn0 12553  ℤ_≥cuz 12903  ...cfz 13567  ..^cfzo 13711  ♯chash 14379   prefix cpfx 14718  Walkscwlks 29632  ClWalkscclwlks 29806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-substr 14689  df-pfx 14719  df-wlks 29635  df-clwlks 29807

This theorem is referenced by:  clwlkclwwlkf1  30042