Theorem reuccatpfxs1 14904

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 2843	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ Word 𝑉 ↔ 𝑦 ∈ Word 𝑉))
2		fveqeq2 6899	. . . 4 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑊) + 1) ↔ (♯‘𝑦) = ((♯‘𝑊) + 1)))
3	1, 2	anbi12d 644	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1)) ↔ (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))))
4	3	cbvralvw 3240	. 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 2940	. . . 4 ⊢ Ⅎ𝑣(𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋
7	5	nfel2 2940	. . . 4 ⊢ Ⅎ𝑣(𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋
8		s1eq 14754	. . . . . 6 ⊢ (𝑣 = 𝑥 → ⟨“𝑣”⟩ = ⟨“𝑥”⟩)
9	8	oveq2d 7440	. . . . 5 ⊢ (𝑣 = 𝑥 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑥”⟩))
10	9	eleq1d 2845	. . . 4 ⊢ (𝑣 = 𝑥 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋))
11		s1eq 14754	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⟨“𝑥”⟩ = ⟨“𝑢”⟩)
12	11	oveq2d 7440	. . . . 5 ⊢ (𝑥 = 𝑢 → (𝑊 ++ ⟨“𝑥”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
13	12	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝑢 → ((𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
14	6, 7, 10, 13	reu8nf 3824	. . 3 ⊢ (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣 ∈ 𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢 ∈ 𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋 → 𝑣 = 𝑢)))
15		nfv 1946	. . . . 5 ⊢ Ⅎ𝑣 𝑊 ∈ Word 𝑉
16		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑣(𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
17	5, 16	nfralw 3309	. . . . 5 ⊢ Ⅎ𝑣∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
18	15, 17	nfan 1931	. . . 4 ⊢ Ⅎ𝑣(𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
19		nfv 1946	. . . . 5 ⊢ Ⅎ𝑣 𝑊 = (𝑥 prefix (♯‘𝑊))
20	5, 19	nfreuw 3395	. . . 4 ⊢ Ⅎ𝑣∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))
21		simprl 783	. . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
23	22	ad2antrr 739	. . . . . . . . . 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 627	. . . . . . . . 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 790	. . . . . . . . 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 796	. . . . . . . . 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 14903	. . . . . . . . 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 1398	. . . . . . . 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 7431	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑥 prefix (♯‘𝑊)) = ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)))
30		s1cl 14756	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝑉 → ⟨“𝑣”⟩ ∈ Word 𝑉)
31	22, 30	anim12i 625	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣 ∈ 𝑉) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
32	31	ad2antrr 739	. . . . . . . . . . . 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 14859	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)) = 𝑊)
34	32, 33	syl 18	. . . . . . . . . . 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 2817	. . . . . . . . . 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 2766	. . . . . . . . 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 418	. . . . . . . 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 215	. . . . . . 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 3154	. . . . . 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 3685	. . . . . 6 ⊢ (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝑊 = (𝑥 prefix (♯‘𝑊)) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊)))
41	21, 39, 40	syl2anc 596	. . . . 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 425	. . . 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 3269	. . 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 245	. 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 606	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Ⅎwnfc 2907  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  ‘cfv 6543  (class class class)co 7424  1c1 11208   + caddc 11210  ♯chash 14481  Word cword 14665   ++ cconcat 14722  ⟨“cs1 14749   prefix cpfx 14828

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-er 8725  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-n0 12614  df-xnn0 12687  df-z 12701  df-uz 12973  df-fz 13647  df-fzo 13796  df-hash 14482  df-word 14666  df-lsw 14715  df-concat 14723  df-s1 14750  df-substr 14797  df-pfx 14829

This theorem is used by:  reuccatpfxs1v  14905  numclwlk2lem2f1o  30989