Theorem scshwfzeqfzo 14391

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 14088	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ ℕ0)
2		elnn0uz 12479	. . . . . . . . . . . 12 ⊢ ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ≥‘0))
3	1, 2	sylib 221	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ (ℤ≥‘0))
4	3	adantr 484	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ (ℤ≥‘0))
5		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ (ℤ≥‘0) ↔ (♯‘𝑋) ∈ (ℤ≥‘0)))
6	5	adantl 485	. . . . . . . . . 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 1133	. . . . . . . 8 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ≥‘0))
9	8	adantr 484	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ≥‘0))
10		fzisfzounsn 13354	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
11	9, 10	syl 17	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (0...𝑁) = ((0..^𝑁) ∪ {𝑁}))
12	11	rexeqdv 3326	. . . . 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 4104	. . . . 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 6730	. . . . . . . . . . . 12 ⊢ (♯‘𝑋) ∈ V
16		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ V ↔ (♯‘𝑋) ∈ V))
17	15, 16	mpbiri 261	. . . . . . . . . . 11 ⊢ (𝑁 = (♯‘𝑋) → 𝑁 ∈ V)
18		oveq2 7221	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift 𝑁))
19	18	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
20	19	rexsng 4590	. . . . . . . . . . 11 ⊢ (𝑁 ∈ V → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
21	17, 20	syl 17	. . . . . . . . . 10 ⊢ (𝑁 = (♯‘𝑋) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
22	21	3ad2ant3 1137	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
23	22	adantr 484	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
24		oveq2 7221	. . . . . . . . . . . . 13 ⊢ (𝑁 = (♯‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
25	24	3ad2ant3 1137	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
26		cshwn 14362	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
27	26	3ad2ant1 1135	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
28	25, 27	eqtrd 2777	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = 𝑋)
29	28	eqeq2d 2748	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
30	29	adantr 484	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
31		cshw0 14359	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
32	31	3ad2ant1 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
33		lennncl 14089	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ)
34	33	3adant3 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ ℕ)
35		eleq1 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
36	35	3ad2ant3 1137	. . . . . . . . . . . . . . . . 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 13282	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
39	37, 38	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 0 ∈ (0..^𝑁))
40		oveq2 7221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 = 𝑛 → (𝑋 cyclShift 0) = (𝑋 cyclShift 𝑛))
41	40	eqeq1d 2739	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 = 𝑛 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
42	41	eqcoms 2745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
43		eqcom 2744	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 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 3528	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → ((𝑋 cyclShift 0) = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛)))
48	32, 47	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
49	48	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
50	49	adantr 484	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
51		eqeq1 2741	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
52	51	adantl 485	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
53	52	rexbidv 3216	. . . . . . . . . . 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 416	. . . . . . . . 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 32	. . . . . 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 858	. . . . 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 32	. . . 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 13261	. . . 4 ⊢ (0..^𝑁) ⊆ (0...𝑁)
63		ssrexv 3968	. . . 4 ⊢ ((0..^𝑁) ⊆ (0...𝑁) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
64	62, 63	mp1i 13	. . 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 3388	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)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∃wrex 3062  {crab 3065  Vcvv 3408   ∪ cun 3864   ⊆ wss 3866  ∅c0 4237  {csn 4541  ‘cfv 6380  (class class class)co 7213  0cc0 10729  ℕcn 11830  ℕ₀cn0 12090  ℤ_≥cuz 12438  ...cfz 13095  ..^cfzo 13238  ♯chash 13896  Word cword 14069   cyclShift ccsh 14353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-inf 9059  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-fl 13367  df-mod 13443  df-hash 13897  df-word 14070  df-concat 14126  df-substr 14206  df-pfx 14236  df-csh 14354

This theorem is referenced by:  hashecclwwlkn1  28160  umgrhashecclwwlk  28161