cshw1 - Metamath Proof Explorer

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Theorem cshw1 14845

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 4493	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 7418	. . . . . 6 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = (0..^0))
3		fzo0 13705	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2787	. . . . 5 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = ∅)
5	4	raleqdv 3309	. . . 4 ⊢ ((♯‘𝑊) = 0 → (∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)))
6	1, 5	mpbiri 258	. . 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 770	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 𝑊 ∈ Word 𝑉)
9		lencl 14556	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
10		1nn0 12522	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2934	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 0 ↔ ¬ (♯‘𝑊) = 0)
13		elnnne0 12520	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) ∈ ℕ ↔ ((♯‘𝑊) ∈ ℕ₀ ∧ (♯‘𝑊) ≠ 0))
14	13	simplbi2com 502	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 0 → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
15	12, 14	sylbir 235	. . . . . . . . . . . . . . 15 ⊢ (¬ (♯‘𝑊) = 0 → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
16	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((♯‘𝑊) ∈ ℕ₀ → (♯‘𝑊) ∈ ℕ))
17	16	impcom 407	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (♯‘𝑊) ∈ ℕ)
18		neqne 2941	. . . . . . . . . . . . . . 15 ⊢ (¬ (♯‘𝑊) = 1 → (♯‘𝑊) ≠ 1)
19	18	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (♯‘𝑊) ≠ 1)
20		nngt1ne1 12274	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 < (♯‘𝑊))
23		elfzo0 13722	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0..^(♯‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 1 < (♯‘𝑊)))
24	11, 17, 22, 23	syl3anbrc 1344	. . . . . . . . . . . 12 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 ∈ (0..^(♯‘𝑊)))
25	24	ex 412	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → 1 ∈ (0..^(♯‘𝑊))))
28	27	impcom 407	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 1 ∈ (0..^(♯‘𝑊)))
29		simprr 772	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (𝑊 cyclShift 1) = 𝑊)
30		lbfzo0 13721	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
31	30, 13	sylbbr 236	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (♯‘𝑊) ≠ 0) → 0 ∈ (0..^(♯‘𝑊)))
32	31	ex 412	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) ∈ ℕ₀ → ((♯‘𝑊) ≠ 0 → 0 ∈ (0..^(♯‘𝑊))))
33	12, 32	biimtrrid 243	. . . . . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(♯‘𝑊))))
38	37	imp 406	. . . . . . . 8 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 0 ∈ (0..^(♯‘𝑊)))
39		elfzoelz 13681	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(♯‘𝑊)) → 1 ∈ ℤ)
40		cshweqrep 14844	. . . . . . . . . 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 593	. . . . . . . . 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 406	. . . . . . . 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 838	. . . . . . 7 ⊢ (((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))))
44		0nn0 12521	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
45		fzossnn0 13712	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0..^(♯‘𝑊)) ⊆ ℕ₀)
46		ssralv 4032	. . . . . . . . 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 2743	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0))
49		elfzoelz 13681	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → 𝑖 ∈ ℤ)
50		zre 12597	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
51		ax-1rid 11204	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
52	50, 51	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
53	52	oveq2d 7426	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
54		zcn 12598	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
55	54	addlidd 11441	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
56	53, 55	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
57	49, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
58	57	oveq1d 7425	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = (𝑖 mod (♯‘𝑊)))
59		zmodidfzoimp 13923	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (𝑖 mod (♯‘𝑊)) = 𝑖)
60	58, 59	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = 𝑖)
61	60	fveqeq2d 6889	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0) ↔ (𝑊‘𝑖) = (𝑊‘0)))
62	61	biimpd 229	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0) → (𝑊‘𝑖) = (𝑊‘0)))
63	48, 62	biimtrid 242	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
64	63	ralimia 3071	. . . . . . . 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 412	. . . . 5 ⊢ ((¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
68	67	impancom 451	. . . 4 ⊢ ((¬ (♯‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (¬ (♯‘𝑊) = 1 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
69		eqid 2736	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
70		c0ex 11234	. . . . . . 7 ⊢ 0 ∈ V
71		fveqeq2 6890	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
72	70, 71	ralsn 4662	. . . . . 6 ⊢ (∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0))
73	69, 72	mpbir 231	. . . . 5 ⊢ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)
74		oveq2 7418	. . . . . . 7 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1))
75		fzo01 13768	. . . . . . 7 ⊢ (0..^1) = {0}
76	74, 75	eqtrdi 2787	. . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = {0})
77	76	raleqdv 3309	. . . . 5 ⊢ ((♯‘𝑊) = 1 → (∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)))
78	73, 77	mpbiri 258	. . . 4 ⊢ ((♯‘𝑊) = 1 → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
79	68, 78	pm2.61d2 181	. . 3 ⊢ ((¬ (♯‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
80	79	ex 412	. 2 ⊢ (¬ (♯‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
81	7, 80	pm2.61i 182	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ⊆ wss 3931 ∅c0 4313 {csn 4606 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 < clt 11274 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ..^cfzo 13676 mod cmo 13891 ♯chash 14353 Word cword 14536 cyclShift ccsh 14811

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 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 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-hash 14354 df-word 14537 df-concat 14594 df-substr 14664 df-pfx 14694 df-csh 14812

This theorem is referenced by: cshw1repsw 14846

W3C validator