Theorem reuccatpfxs1 14896

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 2844	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ Word 𝑉 ↔ 𝑦 ∈ Word 𝑉))
2		fveqeq2 6894	. . . 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 3241	. 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 2941	. . . 4 ⊢ Ⅎ𝑣(𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋
7	5	nfel2 2941	. . . 4 ⊢ Ⅎ𝑣(𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋
8		s1eq 14747	. . . . . 6 ⊢ (𝑣 = 𝑥 → ⟨“𝑣”⟩ = ⟨“𝑥”⟩)
9	8	oveq2d 7436	. . . . 5 ⊢ (𝑣 = 𝑥 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑥”⟩))
10	9	eleq1d 2846	. . . 4 ⊢ (𝑣 = 𝑥 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋))
11		s1eq 14747	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⟨“𝑥”⟩ = ⟨“𝑢”⟩)
12	11	oveq2d 7436	. . . . 5 ⊢ (𝑥 = 𝑢 → (𝑊 ++ ⟨“𝑥”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
13	12	eleq1d 2846	. . . 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 1947	. . . . 5 ⊢ Ⅎ𝑣 𝑊 ∈ Word 𝑉
16		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑣(𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
17	5, 16	nfralw 3310	. . . . 5 ⊢ Ⅎ𝑣∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
18	15, 17	nfan 1932	. . . 4 ⊢ Ⅎ𝑣(𝑊 ∈ Word 𝑉 ∧ ∀𝑦 ∈ 𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
19		nfv 1947	. . . . 5 ⊢ Ⅎ𝑣 𝑊 = (𝑥 prefix (♯‘𝑊))
20	5, 19	nfreuw 3396	. . . 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 14895	. . . . . . . . 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 7427	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑥 prefix (♯‘𝑊)) = ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)))
30		s1cl 14749	. . . . . . . . . . . . . 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 14851	. . . . . . . . . . . 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 2818	. . . . . . . . . 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 2767	. . . . . . . . 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 3155	. . . . . 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 3686	. . . . . 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 3270	. . 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 1570   ∈ wcel 2145  Ⅎwnfc 2908  ∀wral 3077  ∃wrex 3087  ∃!wreu 3364  ‘cfv 6538  (class class class)co 7420  1c1 11201   + caddc 11203  ♯chash 14474  Word cword 14658   ++ cconcat 14715  ⟨“cs1 14742   prefix cpfx 14820

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-xnn0 12680  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-hash 14475  df-word 14659  df-lsw 14708  df-concat 14716  df-s1 14743  df-substr 14789  df-pfx 14821

This theorem is used by:  reuccatpfxs1v  14897  numclwlk2lem2f1o  30980