Theorem scshwfzeqfzo 13572

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 13324	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Word 𝑉 → (#‘𝑋) ∈ ℕ0)
2		elnn0uz 11725	. . . . . . . . . . . 12 ⊢ ((#‘𝑋) ∈ ℕ0 ↔ (#‘𝑋) ∈ (ℤ≥‘0))
3	1, 2	sylib 208	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝑉 → (#‘𝑋) ∈ (ℤ≥‘0))
4	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝑋)) → (#‘𝑋) ∈ (ℤ≥‘0))
5		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑁 = (#‘𝑋) → (𝑁 ∈ (ℤ≥‘0) ↔ (#‘𝑋) ∈ (ℤ≥‘0)))
6	5	adantl 482	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝑋)) → (𝑁 ∈ (ℤ≥‘0) ↔ (#‘𝑋) ∈ (ℤ≥‘0)))
7	4, 6	mpbird 247	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (#‘𝑋)) → 𝑁 ∈ (ℤ≥‘0))
8	7	3adant2 1080	. . . . . . . 8 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 𝑁 ∈ (ℤ≥‘0))
9	8	adantr 481	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ≥‘0))
10		fzisfzounsn 12580	. . . . . . 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 3145	. . . . 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 3793	. . . . 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	syl6bb 276	. . . 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		ax-1 6	. . . . . 6 ⊢ (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
16		fvex 6201	. . . . . . . . . . . 12 ⊢ (#‘𝑋) ∈ V
17		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑁 = (#‘𝑋) → (𝑁 ∈ V ↔ (#‘𝑋) ∈ V))
18	16, 17	mpbiri 248	. . . . . . . . . . 11 ⊢ (𝑁 = (#‘𝑋) → 𝑁 ∈ V)
19		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝑋 cyclShift 𝑛) = (𝑋 cyclShift 𝑁))
20	19	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
21	20	rexsng 4219	. . . . . . . . . . 11 ⊢ (𝑁 ∈ V → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
22	18, 21	syl 17	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝑋) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
23	22	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
24	23	adantr 481	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
25		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑁 = (#‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (#‘𝑋)))
26	25	3ad2ant3 1084	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (#‘𝑋)))
27		cshwn 13543	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (#‘𝑋)) = 𝑋)
28	27	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift (#‘𝑋)) = 𝑋)
29	26, 28	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 𝑁) = 𝑋)
30	29	eqeq2d 2632	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
31	30	adantr 481	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
32		cshw0 13540	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
33	32	3ad2ant1 1082	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
34		lennncl 13325	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅) → (#‘𝑋) ∈ ℕ)
35	34	3adant3 1081	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (#‘𝑋) ∈ ℕ)
36		eleq1 2689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (#‘𝑋) → (𝑁 ∈ ℕ ↔ (#‘𝑋) ∈ ℕ))
37	36	3ad2ant3 1084	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → (𝑁 ∈ ℕ ↔ (#‘𝑋) ∈ ℕ))
38	35, 37	mpbird 247	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 𝑁 ∈ ℕ)
39		lbfzo0 12507	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
40	38, 39	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → 0 ∈ (0..^𝑁))
41		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 = 𝑛 → (𝑋 cyclShift 0) = (𝑋 cyclShift 𝑛))
42	41	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 = 𝑛 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
43	42	eqcoms 2630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ (𝑋 cyclShift 𝑛) = 𝑋))
44		eqcom 2629	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 cyclShift 𝑛) = 𝑋 ↔ 𝑋 = (𝑋 cyclShift 𝑛))
45	43, 44	syl6bb 276	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
46	45	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋 ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
47	46	biimpd 219	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋 → 𝑋 = (𝑋 cyclShift 𝑛)))
48	40, 47	rspcimedv 3311	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → ((𝑋 cyclShift 0) = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛)))
49	33, 48	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
50	49	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
51	50	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
52		eqeq1 2626	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
53	52	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
54	53	rexbidv 3052	. . . . . . . . . . 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)))
55	51, 54	mpbird 247	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛))
56	55	ex 450	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
57	31, 56	sylbid 230	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
58	24, 57	sylbid 230	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
59	58	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)))
60	15, 59	jaoi 394	. . . . 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)))
61	60	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)))
62	14, 61	sylbid 230	. . 3 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
63		fzossfz 12488	. . . 4 ⊢ (0..^𝑁) ⊆ (0...𝑁)
64		ssrexv 3667	. . . 4 ⊢ ((0..^𝑁) ⊆ (0...𝑁) → (∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛) → ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
65	63, 64	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)))
66	62, 65	impbid 202	. 2 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
67	66	rabbidva 3188	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 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  {crab 2916  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  ‘cfv 5888  (class class class)co 6650  0cc0 9936  ℕcn 11020  ℕ₀cn0 11292  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465  #chash 13117  Word cword 13291   cyclShift ccsh 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535

This theorem is referenced by:  hashecclwwlksn1  26954  umgrhashecclwwlk  26955