Theorem clwlkclwwlkf1lem3 27784

Description: Lemma 3 for clwlkclwwlkf1 27788. (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 27783	. . . 4 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))
7		simprr 771	. . . . 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 27782	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
9	1, 4, 5	clwlkclwwlkflem 27782	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
10		lbfzo0 13078	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^(♯‘𝐴)) ↔ (♯‘𝐴) ∈ ℕ)
11	10	biimpri 230	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) ∈ ℕ → 0 ∈ (0..^(♯‘𝐴)))
12	11	3ad2ant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 0 ∈ (0..^(♯‘𝐴)))
13	12	adantr 483	. . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 6670	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝐵‘𝑖) = (𝐵‘0))
16		fveq2 6670	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝐸‘𝑖) = (𝐸‘0))
17	15, 16	eqeq12d 2837	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → ((𝐵‘𝑖) = (𝐸‘𝑖) ↔ (𝐵‘0) = (𝐸‘0)))
18	17	rspcv 3618	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐵‘0))
21		eqtr 2841	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
22	21	adantl 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
23	20, 22	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))
24	23	exp32 423	. . . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
27	26	3ad2ant2 1130	. . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐷) = (♯‘𝐴) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
34	33	eqcoms 2829	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) = (♯‘𝐷) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
35	34	adantl 484	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 2856	. . . . . . . . . . . 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 415	. . . . . . . . . . 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 415	. . . . . . . . 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 597	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((♯‘𝐴) = (♯‘𝐷) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
42	41	impd 413	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
43	42	3adant3 1128	. . . . . 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 409	. . . . 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 514	. . . 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 685	. . 3 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
47		fvex 6683	. . . 4 ⊢ (♯‘𝐴) ∈ V
48		fveq2 6670	. . . . . 6 ⊢ (𝑖 = (♯‘𝐴) → (𝐵‘𝑖) = (𝐵‘(♯‘𝐴)))
49		fveq2 6670	. . . . . 6 ⊢ (𝑖 = (♯‘𝐴) → (𝐸‘𝑖) = (𝐸‘(♯‘𝐴)))
50	48, 49	eqeq12d 2837	. . . . 5 ⊢ (𝑖 = (♯‘𝐴) → ((𝐵‘𝑖) = (𝐸‘𝑖) ↔ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
51	50	ralunsn 4824	. . . 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 236	. 2 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵‘𝑖) = (𝐸‘𝑖))
54		nnnn0 11905	. . . . . . . 8 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
55		elnn0uz 12284	. . . . . . . 8 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ≥‘0))
56	54, 55	sylib 220	. . . . . . 7 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ (ℤ≥‘0))
57	56	3ad2ant3 1131	. . . . . 6 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ (ℤ≥‘0))
58	8, 57	syl 17	. . . . 5 ⊢ (𝑈 ∈ 𝐶 → (♯‘𝐴) ∈ (ℤ≥‘0))
59	58	3ad2ant1 1129	. . . 4 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (♯‘𝐴) ∈ (ℤ≥‘0))
60		fzisfzounsn 13150	. . . 4 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘0) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
61	59, 60	syl 17	. . 3 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
62	61	raleqdv 3415	. 2 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (∀𝑖 ∈ (0...(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖) ↔ ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵‘𝑖) = (𝐸‘𝑖)))
63	53, 62	mpbird 259	1 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {crab 3142  Vcvv 3494   ∪ cun 3934  {csn 4567   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  1^st c1st 7687  2^nd c2nd 7688  0cc0 10537  1c1 10538   ≤ cle 10676  ℕcn 11638  ℕ₀cn0 11898  ℤ_≥cuz 12244  ...cfz 12893  ..^cfzo 13034  ♯chash 13691   prefix cpfx 14032  Walkscwlks 27378  ClWalkscclwlks 27551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-hash 13692  df-word 13863  df-substr 14003  df-pfx 14033  df-wlks 27381  df-clwlks 27552

This theorem is referenced by:  clwlkclwwlkf1  27788