Theorem scshwfzeqfzo 14862

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 14569	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ ℕ0)
2		elnn0uz 12902	. . . . . . . . . . . 12 ⊢ ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ≥‘0))
3	1, 2	sylib 221	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ (ℤ≥‘0))
4	3	adantr 485	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ (ℤ≥‘0))
5		eleq1 2849	. . . . . . . . . . 11 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ (ℤ≥‘0) ↔ (♯‘𝑋) ∈ (ℤ≥‘0)))
6	5	adantl 486	. . . . . . . . . 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 1147	. . . . . . . 8 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ≥‘0))
9	8	adantr 485	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ≥‘0))
10		fzisfzounsn 13808	. . . . . . 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 3322	. . . . 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 4148	. . . . 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 6894	. . . . . . . . . . . 12 ⊢ (♯‘𝑋) ∈ V
16		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ V ↔ (♯‘𝑋) ∈ V))
17	15, 16	mpbiri 261	. . . . . . . . . . 11 ⊢ (𝑁 = (♯‘𝑋) → 𝑁 ∈ V)
18		oveq2 7418	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift 𝑁))
19	18	eqeq2d 2772	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
20	19	rexsng 4641	. . . . . . . . . . 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 1151	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
23	22	adantr 485	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
24		oveq2 7418	. . . . . . . . . . . . 13 ⊢ (𝑁 = (♯‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
25	24	3ad2ant3 1151	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
26		cshwn 14833	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
27	26	3ad2ant1 1149	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
28	25, 27	eqtrd 2796	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = 𝑋)
29	28	eqeq2d 2772	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
30	29	adantr 485	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
31		cshw0 14830	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
32	31	3ad2ant1 1149	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
33		lennncl 14570	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ)
34	33	3adant3 1148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ ℕ)
35		eleq1 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
36	35	3ad2ant3 1151	. . . . . . . . . . . . . . . . 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 13727	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
39	37, 38	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 0 ∈ (0..^𝑁))
40		oveq2 7418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 = 𝑛 → (𝑋 cyclShift 0) = (𝑋 cyclShift 𝑛))
41	40	eqeq1d 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 = 𝑛 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
42	41	eqcoms 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
43		eqcom 2768	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 3571	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
50	49	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
51		eqeq1 2765	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
52	51	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
53	52	rexbidv 3187	. . . . . . . . . . 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 417	. . . . . . . . 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 871	. . . . 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 13706	. . . 4 ⊢ (0..^𝑁) ⊆ (0...𝑁)
63		ssrexv 4006	. . . 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 3420	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 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3087  {crab 3414  Vcvv 3453   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4588  ‘cfv 6536  (class class class)co 7410  0cc0 11099  ℕcn 12232  ℕ₀cn0 12503  ℤ_≥cuz 12861  ...cfz 13534  ..^cfzo 13681  ♯chash 14365  Word cword 14549   cyclShift ccsh 14824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  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 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-inf 9402  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-n0 12504  df-z 12591  df-uz 12862  df-rp 13016  df-fz 13535  df-fzo 13682  df-fl 13824  df-mod 13902  df-hash 14366  df-word 14550  df-concat 14607  df-substr 14678  df-pfx 14708  df-csh 14825

This theorem is referenced by:  hashecclwwlkn1  30394  umgrhashecclwwlk  30395