Theorem clwlkclwwlkf1lem3 30477

Description: Lemma 3 for clwlkclwwlkf1 30481. (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 30476	. . . 4 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))
7		simprr 785	. . . . 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 30475	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
9	1, 4, 5	clwlkclwwlkflem 30475	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
10		lbfzo0 13757	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^(♯‘𝐴)) ↔ (♯‘𝐴) ∈ ℕ)
11	10	biimpri 231	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) ∈ ℕ → 0 ∈ (0..^(♯‘𝐴)))
12	11	3ad2ant3 1153	. . . . . . . . . . . . . 14 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 0 ∈ (0..^(♯‘𝐴)))
13	12	adantr 486	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 6882	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝐵‘𝑖) = (𝐵‘0))
16		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝐸‘𝑖) = (𝐸‘0))
17	15, 16	eqeq12d 2778	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → ((𝐵‘𝑖) = (𝐸‘𝑖) ↔ (𝐵‘0) = (𝐸‘0)))
18	17	rspcv 3575	. . . . . . . . . . . 12 ⊢ (0 ∈ (0..^(♯‘𝐴)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) → (𝐵‘0) = (𝐸‘0)))
19	14, 18	syl 18	. . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐵‘0))
21		eqtr 2782	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
22	21	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
23	20, 22	eqtrd 2797	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))
24	23	exp32 426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵‘(♯‘𝐴)) = (𝐵‘0) → ((𝐵‘0) = (𝐸‘0) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
25	24	com23 87	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵‘(♯‘𝐴)) = (𝐵‘0) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
26	25	eqcoms 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
27	26	3ad2ant2 1152	. . . . . . . . . . . . . . . . . 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 33	. . . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐷) = (♯‘𝐴) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
34	33	eqcoms 2770	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) = (♯‘𝐷) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
35	34	adantl 487	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 2797	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 48	. . . . . . . . . 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 418	. . . . . . . . 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 608	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((♯‘𝐴) = (♯‘𝐷) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
42	41	impd 416	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
43	42	3adant3 1150	. . . . . 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 412	. . . . 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 521	. . . 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 700	. . 3 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
47		fvex 6895	. . . 4 ⊢ (♯‘𝐴) ∈ V
48		fveq2 6882	. . . . . 6 ⊢ (𝑖 = (♯‘𝐴) → (𝐵‘𝑖) = (𝐵‘(♯‘𝐴)))
49		fveq2 6882	. . . . . 6 ⊢ (𝑖 = (♯‘𝐴) → (𝐸‘𝑖) = (𝐸‘(♯‘𝐴)))
50	48, 49	eqeq12d 2778	. . . . 5 ⊢ (𝑖 = (♯‘𝐴) → ((𝐵‘𝑖) = (𝐸‘𝑖) ↔ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
51	50	ralunsn 4857	. . . 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 237	. 2 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵‘𝑖) = (𝐸‘𝑖))
54		nnnn0 12538	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
55		elnn0uz 12931	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ≥‘0))
56	54, 55	sylib 221	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ (ℤ≥‘0))
57	56	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ (ℤ≥‘0))
58	8, 57	syl 18	. . . 4 ⊢ (𝑈 ∈ 𝐶 → (♯‘𝐴) ∈ (ℤ≥‘0))
59	58	3ad2ant1 1151	. . 3 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (♯‘𝐴) ∈ (ℤ≥‘0))
60		fzisfzounsn 13838	. . 3 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘0) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
61	59, 60	syl 18	. 2 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
62	53, 61	raleqtrrdv 3325	1 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453   ∪ cun 3900  {csn 4587   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  1^st c1st 7987  2^nd c2nd 7988  0cc0 11127  1c1 11128   ≤ cle 11271  ℕcn 12260  ℕ₀cn0 12531  ℤ_≥cuz 12890  ...cfz 13563  ..^cfzo 13711  ♯chash 14396   prefix cpfx 14742  Walkscwlks 30057  ClWalkscclwlks 30237

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13564  df-fzo 13712  df-hash 14397  df-word 14581  df-substr 14711  df-pfx 14743  df-wlks 30060  df-clwlks 30238

This theorem is used by:  clwlkclwwlkf1  30481