cshw1 - Metamath Proof Explorer

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Theorem cshw1 14897

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 4457	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 7425	. . . . . 6 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = (0..^0))
3		fzo0 13743	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2813	. . . . 5 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = ∅)
5	4	raleqdv 3321	. . . 4 ⊢ ((♯‘𝑊) = 0 → (∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)))
6	1, 5	mpbiri 261	. . 3 ⊢ ((♯‘𝑊) = 0 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
7	6	a1d 26	. 2 ⊢ ((♯‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
8		simprl 783	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 𝑊 ∈ Word 𝑉)
9		lencl 14602	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
10		1nn0 12548	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2958	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 0 ↔ ¬ (♯‘𝑊) = 0)
13		elnnne0 12546	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) ∈ ℕ ↔ ((♯‘𝑊) ∈ ℕ₀ ∧ (♯‘𝑊) ≠ 0))
14	13	simplbi2com 508	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 0 → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
15	12, 14	sylbir 238	. . . . . . . . . . . . . . 15 ⊢ (¬ (♯‘𝑊) = 0 → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
16	15	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
17	16	impcom 413	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (♯‘𝑊) ∈ ℕ)
18		neqne 2965	. . . . . . . . . . . . . . 15 ⊢ (¬ (♯‘𝑊) = 1 → (♯‘𝑊) ≠ 1)
19	18	ad2antll 742	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (♯‘𝑊) ≠ 1)
20		nngt1ne1 12293	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) ∈ ℕ → (1 < (♯‘𝑊) ↔ (♯‘𝑊) ≠ 1))
21	17, 20	syl 18	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (1 < (♯‘𝑊) ↔ (♯‘𝑊) ≠ 1))
22	19, 21	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 < (♯‘𝑊))
23		elfzo0 13760	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0..^(♯‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 1 < (♯‘𝑊)))
24	11, 17, 22, 23	syl3anbrc 1362	. . . . . . . . . . . 12 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 ∈ (0..^(♯‘𝑊)))
25	24	ex 418	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) ∈ ℕ₀ → ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → 1 ∈ (0..^(♯‘𝑊))))
26	9, 25	syl 18	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → 1 ∈ (0..^(♯‘𝑊))))
27	26	adantr 486	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → 1 ∈ (0..^(♯‘𝑊))))
28	27	impcom 413	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 1 ∈ (0..^(♯‘𝑊)))
29		simprr 785	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (𝑊 cyclShift 1) = 𝑊)
30		lbfzo0 13759	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
31	30, 13	sylbbr 239	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (♯‘𝑊) ≠ 0) → 0 ∈ (0..^(♯‘𝑊)))
32	31	ex 418	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) ∈ ℕ₀ → ((♯‘𝑊) ≠ 0 → 0 ∈ (0..^(♯‘𝑊))))
33	12, 32	biimtrrid 246	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) ∈ ℕ₀ → (¬ (♯‘𝑊) = 0 → 0 ∈ (0..^(♯‘𝑊))))
34	9, 33	syl 18	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → (¬ (♯‘𝑊) = 0 → 0 ∈ (0..^(♯‘𝑊))))
35	34	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → (¬ (♯‘𝑊) = 0 → 0 ∈ (0..^(♯‘𝑊))))
36	35	com12 33	. . . . . . . . . 10 ⊢ (¬ (♯‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(♯‘𝑊))))
37	36	adantr 486	. . . . . . . . 9 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(♯‘𝑊))))
38	37	imp 412	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 0 ∈ (0..^(♯‘𝑊)))
39		elfzoelz 13718	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(♯‘𝑊)) → 1 ∈ ℤ)
40		cshweqrep 14896	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ ℤ) → (((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(♯‘𝑊))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊)))))
41	39, 40	sylan2 605	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘𝑊))) → (((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(♯‘𝑊))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊)))))
42	41	imp 412	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(♯‘𝑊))) ∧ ((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(♯‘𝑊)))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))))
43	8, 28, 29, 38, 42	syl22anc 852	. . . . . . 7 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))))
44		0nn0 12547	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
45		fzossnn0 13750	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0..^(♯‘𝑊)) ⊆ ℕ₀)
46		ssralv 4003	. . . . . . . . 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)))))
47	44, 45, 46	mp2b 10	. . . . . . . 8 ⊢ (∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))))
48		eqcom 2769	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0))
49		elfzoelz 13718	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → 𝑖 ∈ ℤ)
50		zre 12623	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
51		ax-1rid 11198	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
52	50, 51	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
53	52	oveq2d 7433	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
54		zcn 12624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
55	54	addlidd 11439	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
56	53, 55	eqtrd 2797	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
57	49, 56	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
58	57	oveq1d 7432	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = (𝑖 mod (♯‘𝑊)))
59		zmodidfzoimp 13966	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (𝑖 mod (♯‘𝑊)) = 𝑖)
60	58, 59	eqtrd 2797	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = 𝑖)
61	60	fveqeq2d 6890	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0) ↔ (𝑊‘𝑖) = (𝑊‘0)))
62	61	biimpd 232	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0) → (𝑊‘𝑖) = (𝑊‘0)))
63	48, 62	biimtrid 245	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
64	63	ralimia 3098	. . . . . . . 8 ⊢ (∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
65	47, 64	syl 18	. . . . . . 7 ⊢ (∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
66	43, 65	syl 18	. . . . . 6 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
67	66	ex 418	. . . . 5 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
68	67	impancom 457	. . . 4 ⊢ ((¬ (♯‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (¬ (♯‘𝑊) = 1 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
69		eqid 2762	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
70		c0ex 11228	. . . . . . 7 ⊢ 0 ∈ V
71		fveqeq2 6891	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
72	70, 71	ralsn 4645	. . . . . 6 ⊢ (∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0))
73	69, 72	mpbir 234	. . . . 5 ⊢ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)
74		oveq2 7425	. . . . . . 7 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1))
75		fzo01 13807	. . . . . . 7 ⊢ (0..^1) = {0}
76	74, 75	eqtrdi 2813	. . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = {0})
77	76	raleqdv 3321	. . . . 5 ⊢ ((♯‘𝑊) = 1 → (∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)))
78	73, 77	mpbiri 261	. . . 4 ⊢ ((♯‘𝑊) = 1 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
79	68, 78	pm2.61d2 183	. . 3 ⊢ ((¬ (♯‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
80	79	ex 418	. 2 ⊢ (¬ (♯‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
81	7, 80	pm2.61i 184	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ⊆ wss 3902 ∅c0 4282 {csn 4587 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 ℝcr 11127 0cc0 11128 1c1 11129 + caddc 11131 · cmul 11133 < clt 11271 ℕcn 12261 ℕ₀cn0 12532 ℤcz 12619 ..^cfzo 13713 mod cmo 13934 ♯chash 14398 Word cword 14582 cyclShift ccsh 14863

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-inf 9417 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-n0 12533 df-z 12620 df-uz 12892 df-rp 13047 df-fz 13566 df-fzo 13714 df-fl 13857 df-mod 13935 df-hash 14399 df-word 14583 df-concat 14640 df-substr 14713 df-pfx 14745 df-csh 14864

This theorem is used by: cshw1repsw 14898

W3C validator