Theorem reuccatpfxs1 14715

Description: There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 13-Oct-2022.)

Hypothesis

Ref	Expression
reuccatpfxs1.1	⊢ Ⅎ𝑣𝑋

Assertion

Ref	Expression
reuccatpfxs1	⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))

Distinct variable groups:   𝑣,𝑉,𝑥   𝑣,𝑊,𝑥   𝑥,𝑋

Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem reuccatpfxs1

Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2811	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ Word 𝑉 ↔ 𝑦 ∈ Word 𝑉))
2		fveqeq2 6900	. . . 4 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑊) + 1) ↔ (♯‘𝑦) = ((♯‘𝑊) + 1)))
3	1, 2	anbi12d 630	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1)) ↔ (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))))
4	3	cbvralvw 3229	. 2 ⊢ (∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1)) ↔ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
5		reuccatpfxs1.1	. . . . 5 ⊢ Ⅎ𝑣𝑋
6	5	nfel2 2916	. . . 4 ⊢ Ⅎ𝑣(𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋
7	5	nfel2 2916	. . . 4 ⊢ Ⅎ𝑣(𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋
8		s1eq 14568	. . . . . 6 ⊢ (𝑣 = 𝑥 → ⟨“𝑣”⟩ = ⟨“𝑥”⟩)
9	8	oveq2d 7430	. . . . 5 ⊢ (𝑣 = 𝑥 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑥”⟩))
10	9	eleq1d 2813	. . . 4 ⊢ (𝑣 = 𝑥 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋))
11		s1eq 14568	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⟨“𝑥”⟩ = ⟨“𝑢”⟩)
12	11	oveq2d 7430	. . . . 5 ⊢ (𝑥 = 𝑢 → (𝑊 ++ ⟨“𝑥”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
13	12	eleq1d 2813	. . . 4 ⊢ (𝑥 = 𝑢 → ((𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
14	6, 7, 10, 13	reu8nf 3867	. . 3 ⊢ (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣 ∈ 𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)))
15		nfv 1910	. . . . 5 ⊢ Ⅎ𝑣 𝑊 ∈ Word 𝑉
16		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑣(𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
17	5, 16	nfralw 3303	. . . . 5 ⊢ Ⅎ𝑣∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
18	15, 17	nfan 1895	. . . 4 ⊢ Ⅎ𝑣(𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
19		nfv 1910	. . . . 5 ⊢ Ⅎ𝑣 𝑊 = (𝑥 prefix (♯‘𝑊))
20	5, 19	nfreuw 3405	. . . 4 ⊢ Ⅎ𝑣∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))
21		simprl 770	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋)
22		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
23	22	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → 𝑊 ∈ Word 𝑉)
24	23	anim1i 614	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑊 ∈ Word 𝑉 ∧ 𝑥 ∈ 𝑋))
25		simplrr 777	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))
26		simp-4r 783	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
27		reuccatpfxs1lem 14714	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑥 ∈ 𝑋) ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢) ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (𝑊 = (𝑥 prefix (♯‘𝑊)) → 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
28	24, 25, 26, 27	syl3anc 1369	. . . . . . . 8 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑊 = (𝑥 prefix (♯‘𝑊)) → 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
29		oveq1 7421	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑥 prefix (♯‘𝑊)) = ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)))
30		s1cl 14570	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝑉 → ⟨“𝑣”⟩ ∈ Word 𝑉)
31	22, 30	anim12i 612	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
32	31	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
33		pfxccat1 14670	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)) = 𝑊)
34	32, 33	syl 17	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)) = 𝑊)
35	29, 34	sylan9eqr 2789	. . . . . . . . . 10 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑥 prefix (♯‘𝑊)) = 𝑊)
36	35	eqcomd 2733	. . . . . . . . 9 ⊢ ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑊 = (𝑥 prefix (♯‘𝑊)))
37	36	ex 412	. . . . . . . 8 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → 𝑊 = (𝑥 prefix (♯‘𝑊))))
38	28, 37	impbid 211	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑊 = (𝑥 prefix (♯‘𝑊)) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
39	38	ralrimiva 3141	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → ∀𝑥 ∈ 𝑋 (𝑊 = (𝑥 prefix (♯‘𝑊)) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
40		reu6i 3721	. . . . . 6 ⊢ (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑊 = (𝑥 prefix (♯‘𝑊)) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊)))
41	21, 39, 40	syl2anc 583	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢))) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊)))
42	41	exp31 419	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (𝑣 ∈ 𝑉 → (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊)))))
43	18, 20, 42	rexlimd 3258	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (∃𝑣 ∈ 𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
44	14, 43	biimtrid 241	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
45	4, 44	sylan2b 593	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Ⅎwnfc 2878  ∀wral 3056  ∃wrex 3065  ∃!wreu 3369  ‘cfv 6542  (class class class)co 7414  1c1 11125   + caddc 11127  ♯chash 14307  Word cword 14482   ++ cconcat 14538  ⟨“cs1 14563   prefix cpfx 14638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-n0 12489  df-xnn0 12561  df-z 12575  df-uz 12839  df-fz 13503  df-fzo 13646  df-hash 14308  df-word 14483  df-lsw 14531  df-concat 14539  df-s1 14564  df-substr 14609  df-pfx 14639

This theorem is referenced by:  reuccatpfxs1v  14716  numclwlk2lem2f1o  30163