Theorem scshwfzeqfzo 14871

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 14577	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ ℕ0)
2		elnn0uz 12930	. . . . . . . . . . . 12 ⊢ ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ≥‘0))
3	1, 2	sylib 218	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝑉 → (♯‘𝑋) ∈ (ℤ≥‘0))
4	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ (ℤ≥‘0))
5		eleq1 2829	. . . . . . . . . . 11 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ (ℤ≥‘0) ↔ (♯‘𝑋) ∈ (ℤ≥‘0)))
6	5	adantl 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑋)) → (𝑁 ∈ (ℤ≥‘0) ↔ (♯‘𝑋) ∈ (ℤ≥‘0)))
7	4, 6	mpbird 257	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ≥‘0))
8	7	3adant2 1132	. . . . . . . 8 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ (ℤ≥‘0))
9	8	adantr 480	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → 𝑁 ∈ (ℤ≥‘0))
10		fzisfzounsn 13824	. . . . . . 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 3327	. . . . 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 4209	. . . . 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 287	. . . 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 6927	. . . . . . . . . . . 12 ⊢ (♯‘𝑋) ∈ V
16		eleq1 2829	. . . . . . . . . . . 12 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ V ↔ (♯‘𝑋) ∈ V))
17	15, 16	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑁 = (♯‘𝑋) → 𝑁 ∈ V)
18		oveq2 7446	. . . . . . . . . . . . 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 4684	. . . . . . . . . . 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 1136	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
23	22	adantr 480	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (∃𝑛 ∈ {𝑁}𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑦 = (𝑋 cyclShift 𝑁)))
24		oveq2 7446	. . . . . . . . . . . . 13 ⊢ (𝑁 = (♯‘𝑋) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
25	24	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 𝑁) = (𝑋 cyclShift (♯‘𝑋)))
26		cshwn 14841	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift (♯‘𝑋)) = 𝑋)
27	26	3ad2ant1 1134	. . . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = (𝑋 cyclShift 𝑁) ↔ 𝑦 = 𝑋))
31		cshw0 14838	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ Word 𝑉 → (𝑋 cyclShift 0) = 𝑋)
32	31	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑋 cyclShift 0) = 𝑋)
33		lennncl 14578	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅) → (♯‘𝑋) ∈ ℕ)
34	33	3adant3 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (♯‘𝑋) ∈ ℕ)
35		eleq1 2829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑋) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
36	35	3ad2ant3 1136	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → (𝑁 ∈ ℕ ↔ (♯‘𝑋) ∈ ℕ))
37	34, 36	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 𝑁 ∈ ℕ)
38		lbfzo0 13745	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
39	37, 38	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → 0 ∈ (0..^𝑁))
40		oveq2 7446	. . . . . . . . . . . . . . . . . . . 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 287	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → ((𝑋 cyclShift 0) = 𝑋 ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
45	44	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋 ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
46	45	biimpd 229	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑛 = 0) → ((𝑋 cyclShift 0) = 𝑋 → 𝑋 = (𝑋 cyclShift 𝑛)))
47	39, 46	rspcimedv 3616	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → ∃𝑛 ∈ (0..^𝑁)𝑋 = (𝑋 cyclShift 𝑛))
50	49	adantr 480	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → (𝑦 = (𝑋 cyclShift 𝑛) ↔ 𝑋 = (𝑋 cyclShift 𝑛)))
53	52	rexbidv 3179	. . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) ∧ 𝑦 = 𝑋) → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛))
55	54	ex 412	. . . . . . . . 9 ⊢ (((𝑋 ∈ Word 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) ∧ 𝑦 ∈ Word 𝑉) → (𝑦 = 𝑋 → ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)))
56	30, 55	sylbid 240	. . . . . . . 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 240	. . . . . . 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 859	. . . . 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 240	. . 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 13724	. . . 4 ⊢ (0..^𝑁) ⊆ (0...𝑁)
63		ssrexv 4068	. . . 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 212	. 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 3443	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 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  {crab 3436  Vcvv 3481   ∪ cun 3964   ⊆ wss 3966  ∅c0 4342  {csn 4634  ‘cfv 6569  (class class class)co 7438  0cc0 11162  ℕcn 12273  ℕ₀cn0 12533  ℤ_≥cuz 12885  ...cfz 13553  ..^cfzo 13700  ♯chash 14375  Word cword 14558   cyclShift ccsh 14832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761  ax-cnex 11218  ax-resscn 11219  ax-1cn 11220  ax-icn 11221  ax-addcl 11222  ax-addrcl 11223  ax-mulcl 11224  ax-mulrcl 11225  ax-mulcom 11226  ax-addass 11227  ax-mulass 11228  ax-distr 11229  ax-i2m1 11230  ax-1ne0 11231  ax-1rid 11232  ax-rnegex 11233  ax-rrecex 11234  ax-cnre 11235  ax-pre-lttri 11236  ax-pre-lttrn 11237  ax-pre-ltadd 11238  ax-pre-mulgt0 11239  ax-pre-sup 11240

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-pss 3986  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-int 4955  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-tr 5269  df-id 5587  df-eprel 5593  df-po 5601  df-so 5602  df-fr 5645  df-we 5647  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-pred 6329  df-ord 6395  df-on 6396  df-lim 6397  df-suc 6398  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-om 7895  df-1st 8022  df-2nd 8023  df-frecs 8314  df-wrecs 8345  df-recs 8419  df-rdg 8458  df-1o 8514  df-er 8753  df-en 8994  df-dom 8995  df-sdom 8996  df-fin 8997  df-sup 9489  df-inf 9490  df-card 9986  df-pnf 11304  df-mnf 11305  df-xr 11306  df-ltxr 11307  df-le 11308  df-sub 11501  df-neg 11502  df-div 11928  df-nn 12274  df-n0 12534  df-z 12621  df-uz 12886  df-rp 13042  df-fz 13554  df-fzo 13701  df-fl 13838  df-mod 13916  df-hash 14376  df-word 14559  df-concat 14615  df-substr 14685  df-pfx 14715  df-csh 14833

This theorem is referenced by:  hashecclwwlkn1  30122  umgrhashecclwwlk  30123