Theorem scshwfzeqfzo 14977

Description: For a nonempty word the sets of shifted words, expressd by a finite interval of integers or by a half-open integer range are identical. (Contributed by Alexander van der Vekens, 15-Jun-2018.)

Assertion

Ref	Expression
scshwfzeqfzo	⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)})

Distinct variable groups:   𝑛,𝑁,𝑦   𝑛,𝑉,𝑦   𝑛,𝑋,𝑦

Proof of Theorem scshwfzeqfzo

Step	Hyp	Ref	Expression
1		lencl 14678	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ ℕ0)
2		elnn0uz 13006	. . . . . . . . . . . 12 ⊢ ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ≥‘0))
3	1, 2	sylib 221	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ (ℤ≥‘0))
4	3	adantr 486	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ (ℤ≥‘0))
5		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ (ℤ≥‘0) ↔ (♯‘𝑋) ∈ (ℤ≥‘0)))
6	5	adantl 487	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑋)) → (𝑁 ∈ (ℤ≥‘0) ↔ (♯‘𝑋) ∈ (ℤ≥‘0)))
7	4, 6	mpbird 260	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ≥‘0))
8	7	3adant2 1149	. . . . . . . 8 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ≥‘0))
9	8	adantr 486	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ≥‘0))
10		fzisfzounsn 13915	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
11	9, 10	syl 18	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
12	11	rexeqdv 3320	. . . . 5 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ ((0..^𝑁) ∪ {𝑁})𝑦 = (𝑋 cyclShift 𝑛)))
13		rexun 4194	. . . . 5 ⊢ (∃𝑛 ∈ ((0..^𝑁) ∪ {𝑁})𝑦 = (𝑋 cyclShift 𝑛) ↔ (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)))
14	12, 13	bitrdi 290	. . . 4 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛))))
15		fvex 6898	. . . . . . . . . . . 12 ⊢ (♯‘𝑋) ∈ V
16		eleq1 2848	. . . . . . . . . . . 12 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ V ↔ (♯‘𝑋) ∈ V))
17	15, 16	mpbiri 261	. . . . . . . . . . 11 ⊢ (𝑁 = (♯‘𝑋) → 𝑁 ∈ V)
18		oveq2 7428	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift 𝑁))
19	18	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
20	19	rexsng 4636	. . . . . . . . . . 11 ⊢ (𝑁 ∈ V → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
21	17, 20	syl 18	. . . . . . . . . 10 ⊢ (𝑁 = (♯‘𝑋) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
22	21	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
23	22	adantr 486	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
24		oveq2 7428	. . . . . . . . . . . . 13 ⊢ (𝑁 = (♯‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
25	24	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
26		cshwn 14948	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
27	26	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
28	25, 27	eqtrd 2795	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = 𝑋)
29	28	eqeq2d 2771	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
30	29	adantr 486	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
31		cshw0 14945	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
32	31	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
33		lennncl 14679	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ)
34	33	3adant3 1150	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ ℕ)
35		eleq1 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
36	35	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
37	34, 36	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ ℕ)
38		lbfzo0 13834	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
39	37, 38	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 0 ∈ (0..^𝑁))
40		oveq2 7428	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 = 𝑛 → (𝑋 cyclShift 0) = (𝑋 cyclShift 𝑛))
41	40	eqeq1d 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 = 𝑛 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
42	41	eqcoms 2768	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
43		eqcom 2767	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 cyclShift 𝑛) = 𝑋 ↔ 𝑋 = (𝑋 cyclShift 𝑛))
44	42, 43	bitrdi 290	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
45	44	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋 ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
46	45	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋 → 𝑋 = (𝑋 cyclShift 𝑛)))
47	39, 46	rspcimedv 3567	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → ((𝑋 cyclShift 0) = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛)))
48	32, 47	mpd 16	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
49	48	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
50	49	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
51		eqeq1 2764	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
52	51	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
53	52	rexbidv 3186	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛)))
54	50, 53	mpbird 260	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛))
55	54	ex 418	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
56	30, 55	sylbid 243	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
57	23, 56	sylbid 243	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
58	57	com12 33	. . . . . 6 ⊢ (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) → (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
59	58	jao1i 872	. . . . 5 ⊢ ((∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)) → (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
60	59	com12 33	. . . 4 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ((∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) ∨ ∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛)) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
61	14, 60	sylbid 243	. . 3 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
62		fzossfz 13813	. . . 4 ⊢ (0..^𝑁) ⊆ (0...𝑁)
63		ssrexv 3978	. . . 4 ⊢ ((0..^𝑁) ⊆ (0...𝑁) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
64	62, 63	mp1i 14	. . 3 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
65	61, 64	impbid 215	. 2 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
66	65	rabbidva 3418	1 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4583  ‘cfv 6538  (class class class)co 7420  0cc0 11200  ℕcn 12335  ℕ₀cn0 12606  ℤ_≥cuz 12965  ...cfz 13639  ..^cfzo 13788  ♯chash 14474  Word cword 14658   cyclShift ccsh 14939

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 5326  ax-pr 5390  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  ax-pre-sup 11278

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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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-sup 9434  df-inf 9435  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-div 11974  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fz 13640  df-fzo 13789  df-fl 13932  df-mod 14010  df-hash 14475  df-word 14659  df-concat 14716  df-substr 14789  df-pfx 14821  df-csh 14940

This theorem is used by:  hashecclwwlkn1  30668  umgrhashecclwwlk  30669