cshw1 - Metamath Proof Explorer

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Theorem cshw1 14726

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 4463	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 7354	. . . . . 6 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = (0..^0))
3		fzo0 13580	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2782	. . . . 5 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = ∅)
5	4	raleqdv 3292	. . . 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 14437	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
10		1nn0 12394	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2929	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 0 ↔ ¬ (♯‘𝑊) = 0)
13		elnnne0 12392	. . . . . . . . . . . . . . . . 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 2936	. . . . . . . . . . . . . . 15 ⊢ (¬ (♯‘𝑊) = 1 → (♯‘𝑊) ≠ 1)
19	18	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (♯‘𝑊) ≠ 1)
20		nngt1ne1 12151	. . . . . . . . . . . . . . 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 13597	. . . . . . . . . . . . 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 13596	. . . . . . . . . . . . . . . 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 13556	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(♯‘𝑊)) → 1 ∈ ℤ)
40		cshweqrep 14725	. . . . . . . . . 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 12393	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
45		fzossnn0 13587	. . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0))
49		elfzoelz 13556	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → 𝑖 ∈ ℤ)
50		zre 12469	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
51		ax-1rid 11073	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
52	50, 51	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
53	52	oveq2d 7362	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
54		zcn 12470	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
55	54	addlidd 11311	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
56	53, 55	eqtrd 2766	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
57	49, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
58	57	oveq1d 7361	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = (𝑖 mod (♯‘𝑊)))
59		zmodidfzoimp 13802	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (𝑖 mod (♯‘𝑊)) = 𝑖)
60	58, 59	eqtrd 2766	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = 𝑖)
61	60	fveqeq2d 6830	. . . . . . . . . . 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 3066	. . . . . . . 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 2731	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
70		c0ex 11103	. . . . . . 7 ⊢ 0 ∈ V
71		fveqeq2 6831	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
72	70, 71	ralsn 4634	. . . . . 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 7354	. . . . . . 7 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1))
75		fzo01 13644	. . . . . . 7 ⊢ (0..^1) = {0}
76	74, 75	eqtrdi 2782	. . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = {0})
77	76	raleqdv 3292	. . . . 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 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ⊆ wss 3902 ∅c0 4283 {csn 4576 class class class wbr 5091 ‘cfv 6481 (class class class)co 7346 ℝcr 11002 0cc0 11003 1c1 11004 + caddc 11006 · cmul 11008 < clt 11143 ℕcn 12122 ℕ₀cn0 12378 ℤcz 12465 ..^cfzo 13551 mod cmo 13770 ♯chash 14234 Word cword 14417 cyclShift ccsh 14692

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-n0 12379 df-z 12466 df-uz 12730 df-rp 12888 df-fz 13405 df-fzo 13552 df-fl 13693 df-mod 13771 df-hash 14235 df-word 14418 df-concat 14475 df-substr 14546 df-pfx 14576 df-csh 14693

This theorem is referenced by: cshw1repsw 14727

W3C validator