cshw1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cshw1		Structured version Visualization version GIF version

Theorem cshw1 13568

Description: If cyclically shifting a word by 1 position results in the word itself, the word is build of identical symbols. Remark: also "valid" for an empty word! (Contributed by AV, 13-May-2018.) (Revised by AV, 7-Jun-2018.) (Proof shortened by AV, 1-Nov-2018.)

**Assertion**
Ref	Expression
cshw1	⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))

Distinct variable groups: 𝑖,𝑉 𝑖,𝑊

**Proof of Theorem cshw1**
Step	Hyp	Ref	Expression
1		ral0 4076	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 6658	. . . . . 6 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = (0..^0))
3		fzo0 12492	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	syl6eq 2672	. . . . 5 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = ∅)
5	4	raleqdv 3144	. . . 4 ⊢ ((#‘𝑊) = 0 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)))
6	1, 5	mpbiri 248	. . 3 ⊢ ((#‘𝑊) = 0 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
7	6	a1d 25	. 2 ⊢ ((#‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
8		simprl 794	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 𝑊 ∈ Word 𝑉)
9		lencl 13324	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
10		1nn0 11308	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2795	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 0 ↔ ¬ (#‘𝑊) = 0)
13		elnnne0 11306	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℕ₀ ∧ (#‘𝑊) ≠ 0))
14	13	simplbi2com 657	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 0 → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
15	12, 14	sylbir 225	. . . . . . . . . . . . . . 15 ⊢ (¬ (#‘𝑊) = 0 → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
16	15	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
17	16	impcom 446	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (#‘𝑊) ∈ ℕ)
18		df-ne 2795	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 1 ↔ ¬ (#‘𝑊) = 1)
19	18	biimpri 218	. . . . . . . . . . . . . . 15 ⊢ (¬ (#‘𝑊) = 1 → (#‘𝑊) ≠ 1)
20	19	ad2antll 765	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (#‘𝑊) ≠ 1)
21		nngt1ne1 11047	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) ∈ ℕ → (1 < (#‘𝑊) ↔ (#‘𝑊) ≠ 1))
22	17, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (1 < (#‘𝑊) ↔ (#‘𝑊) ≠ 1))
23	20, 22	mpbird 247	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 < (#‘𝑊))
24		elfzo0 12508	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0..^(#‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 1 < (#‘𝑊)))
25	11, 17, 23, 24	syl3anbrc 1246	. . . . . . . . . . . 12 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 ∈ (0..^(#‘𝑊)))
26	25	ex 450	. . . . . . . . . . 11 ⊢ ((#‘𝑊) ∈ ℕ₀ → ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → 1 ∈ (0..^(#‘𝑊))))
27	9, 26	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → 1 ∈ (0..^(#‘𝑊))))
28	27	adantr 481	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → 1 ∈ (0..^(#‘𝑊))))
29	28	impcom 446	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 1 ∈ (0..^(#‘𝑊)))
30		simprr 796	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (𝑊 cyclShift 1) = 𝑊)
31		lbfzo0 12507	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
32	31	biimpri 218	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ∈ ℕ → 0 ∈ (0..^(#‘𝑊)))
33	13, 32	sylbir 225	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (#‘𝑊) ≠ 0) → 0 ∈ (0..^(#‘𝑊)))
34	33	ex 450	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑊) ∈ ℕ₀ → ((#‘𝑊) ≠ 0 → 0 ∈ (0..^(#‘𝑊))))
35	12, 34	syl5bir 233	. . . . . . . . . . . . 13 ⊢ ((#‘𝑊) ∈ ℕ₀ → (¬ (#‘𝑊) = 0 → 0 ∈ (0..^(#‘𝑊))))
36	9, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → (¬ (#‘𝑊) = 0 → 0 ∈ (0..^(#‘𝑊))))
37	36	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → (¬ (#‘𝑊) = 0 → 0 ∈ (0..^(#‘𝑊))))
38	37	com12 32	. . . . . . . . . 10 ⊢ (¬ (#‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(#‘𝑊))))
39	38	adantr 481	. . . . . . . . 9 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(#‘𝑊))))
40	39	imp 445	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 0 ∈ (0..^(#‘𝑊)))
41		elfzoelz 12470	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(#‘𝑊)) → 1 ∈ ℤ)
42		cshweqrep 13567	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ ℤ) → (((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(#‘𝑊))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊)))))
43	41, 42	sylan2 491	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(#‘𝑊))) → (((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(#‘𝑊))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊)))))
44	43	imp 445	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(#‘𝑊))) ∧ ((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(#‘𝑊)))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))))
45	8, 29, 30, 40, 44	syl22anc 1327	. . . . . . 7 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))))
46		0nn0 11307	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
47		fzossnn0 12499	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0..^(#‘𝑊)) ⊆ ℕ₀)
48		ssralv 3666	. . . . . . . . 9 ⊢ ((0..^(#‘𝑊)) ⊆ ℕ₀ → (∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊)))))
49	46, 47, 48	mp2b 10	. . . . . . . 8 ⊢ (∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))))
50		eqcom 2629	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0))
51		elfzoelz 12470	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → 𝑖 ∈ ℤ)
52		zre 11381	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
53		ax-1rid 10006	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
55	54	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
56		zcn 11382	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
57	56	addid2d 10237	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
58	55, 57	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
59	51, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
60	59	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((0 + (𝑖 · 1)) mod (#‘𝑊)) = (𝑖 mod (#‘𝑊)))
61		zmodidfzoimp 12700	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (𝑖 mod (#‘𝑊)) = 𝑖)
62	60, 61	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((0 + (𝑖 · 1)) mod (#‘𝑊)) = 𝑖)
63	62	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘𝑖))
64	63	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0) ↔ (𝑊‘𝑖) = (𝑊‘0)))
65	64	biimpd 219	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0) → (𝑊‘𝑖) = (𝑊‘0)))
66	50, 65	syl5bi 232	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
67	66	ralimia 2950	. . . . . . . 8 ⊢ (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
68	49, 67	syl 17	. . . . . . 7 ⊢ (∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
69	45, 68	syl 17	. . . . . 6 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
70	69	ex 450	. . . . 5 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
71	70	impancom 456	. . . 4 ⊢ ((¬ (#‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (¬ (#‘𝑊) = 1 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
72		eqid 2622	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
73		c0ex 10034	. . . . . . 7 ⊢ 0 ∈ V
74		fveq2 6191	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑊‘𝑖) = (𝑊‘0))
75	74	eqeq1d 2624	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
76	73, 75	ralsn 4222	. . . . . 6 ⊢ (∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0))
77	72, 76	mpbir 221	. . . . 5 ⊢ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)
78		oveq2 6658	. . . . . . 7 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = (0..^1))
79		fzo01 12550	. . . . . . 7 ⊢ (0..^1) = {0}
80	78, 79	syl6eq 2672	. . . . . 6 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = {0})
81	80	raleqdv 3144	. . . . 5 ⊢ ((#‘𝑊) = 1 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)))
82	77, 81	mpbiri 248	. . . 4 ⊢ ((#‘𝑊) = 1 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
83	71, 82	pm2.61d2 172	. . 3 ⊢ ((¬ (#‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
84	83	ex 450	. 2 ⊢ (¬ (#‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
85	7, 84	pm2.61i 176	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 {csn 4177 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ..^cfzo 12465 mod cmo 12668 #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-2 11079 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: cshw1repsw 13569

W3C validator