cshw1 - Metamath Proof Explorer

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Theorem cshw1 14387

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 4424	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 7221	. . . . . 6 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = (0..^0))
3		fzo0 13266	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2794	. . . . 5 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = ∅)
5	4	raleqdv 3325	. . . 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 25	. 2 ⊢ ((♯‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
8		simprl 771	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 𝑊 ∈ Word 𝑉)
9		lencl 14088	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
10		1nn0 12106	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2941	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 0 ↔ ¬ (♯‘𝑊) = 0)
13		elnnne0 12104	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) ∈ ℕ ↔ ((♯‘𝑊) ∈ ℕ₀ ∧ (♯‘𝑊) ≠ 0))
14	13	simplbi2com 506	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 0 → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
15	12, 14	sylbir 238	. . . . . . . . . . . . . . 15 ⊢ (¬ (♯‘𝑊) = 0 → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
16	15	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
17	16	impcom 411	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (♯‘𝑊) ∈ ℕ)
18		neqne 2948	. . . . . . . . . . . . . . 15 ⊢ (¬ (♯‘𝑊) = 1 → (♯‘𝑊) ≠ 1)
19	18	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (♯‘𝑊) ≠ 1)
20		nngt1ne1 11859	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) ∈ ℕ → (1 < (♯‘𝑊) ↔ (♯‘𝑊) ≠ 1))
21	17, 20	syl 17	. . . . . . . . . . . . . 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 13283	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0..^(♯‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 1 < (♯‘𝑊)))
24	11, 17, 22, 23	syl3anbrc 1345	. . . . . . . . . . . 12 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 ∈ (0..^(♯‘𝑊)))
25	24	ex 416	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) ∈ ℕ₀ → ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → 1 ∈ (0..^(♯‘𝑊))))
26	9, 25	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → 1 ∈ (0..^(♯‘𝑊))))
27	26	adantr 484	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → 1 ∈ (0..^(♯‘𝑊))))
28	27	impcom 411	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 1 ∈ (0..^(♯‘𝑊)))
29		simprr 773	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (𝑊 cyclShift 1) = 𝑊)
30		lbfzo0 13282	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
31	30, 13	sylbbr 239	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (♯‘𝑊) ≠ 0) → 0 ∈ (0..^(♯‘𝑊)))
32	31	ex 416	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) ∈ ℕ₀ → ((♯‘𝑊) ≠ 0 → 0 ∈ (0..^(♯‘𝑊))))
33	12, 32	syl5bir 246	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) ∈ ℕ₀ → (¬ (♯‘𝑊) = 0 → 0 ∈ (0..^(♯‘𝑊))))
34	9, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → (¬ (♯‘𝑊) = 0 → 0 ∈ (0..^(♯‘𝑊))))
35	34	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → (¬ (♯‘𝑊) = 0 → 0 ∈ (0..^(♯‘𝑊))))
36	35	com12 32	. . . . . . . . . 10 ⊢ (¬ (♯‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(♯‘𝑊))))
37	36	adantr 484	. . . . . . . . 9 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(♯‘𝑊))))
38	37	imp 410	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 0 ∈ (0..^(♯‘𝑊)))
39		elfzoelz 13243	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(♯‘𝑊)) → 1 ∈ ℤ)
40		cshweqrep 14386	. . . . . . . . . 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 596	. . . . . . . . 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 410	. . . . . . . 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 839	. . . . . . 7 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))))
44		0nn0 12105	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
45		fzossnn0 13273	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0..^(♯‘𝑊)) ⊆ ℕ₀)
46		ssralv 3967	. . . . . . . . 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 2744	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0))
49		elfzoelz 13243	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → 𝑖 ∈ ℤ)
50		zre 12180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
51		ax-1rid 10799	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
52	50, 51	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
53	52	oveq2d 7229	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
54		zcn 12181	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
55	54	addid2d 11033	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
56	53, 55	eqtrd 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
57	49, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
58	57	oveq1d 7228	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = (𝑖 mod (♯‘𝑊)))
59		zmodidfzoimp 13474	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (𝑖 mod (♯‘𝑊)) = 𝑖)
60	58, 59	eqtrd 2777	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = 𝑖)
61	60	fveqeq2d 6725	. . . . . . . . . . 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	syl5bi 245	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
64	63	ralimia 3081	. . . . . . . 8 ⊢ (∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
65	47, 64	syl 17	. . . . . . 7 ⊢ (∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
66	43, 65	syl 17	. . . . . 6 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
67	66	ex 416	. . . . 5 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
68	67	impancom 455	. . . 4 ⊢ ((¬ (♯‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (¬ (♯‘𝑊) = 1 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
69		eqid 2737	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
70		c0ex 10827	. . . . . . 7 ⊢ 0 ∈ V
71		fveqeq2 6726	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
72	70, 71	ralsn 4597	. . . . . 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 7221	. . . . . . 7 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1))
75		fzo01 13324	. . . . . . 7 ⊢ (0..^1) = {0}
76	74, 75	eqtrdi 2794	. . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = {0})
77	76	raleqdv 3325	. . . . 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 184	. . 3 ⊢ ((¬ (♯‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
80	79	ex 416	. 2 ⊢ (¬ (♯‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
81	7, 80	pm2.61i 185	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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ⊆ wss 3866 ∅c0 4237 {csn 4541 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 < clt 10867 ℕcn 11830 ℕ₀cn0 12090 ℤcz 12176 ..^cfzo 13238 mod cmo 13442 ♯chash 13896 Word cword 14069 cyclShift ccsh 14353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-hash 13897 df-word 14070 df-concat 14126 df-substr 14206 df-pfx 14236 df-csh 14354

This theorem is referenced by: cshw1repsw 14388

W3C validator