cshw1 - Metamath Proof Explorer

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Theorem cshw1 14870

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 4536	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 7456	. . . . . 6 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = (0..^0))
3		fzo0 13740	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	eqtrdi 2796	. . . . 5 ⊢ ((♯‘𝑊) = 0 → (0..^(♯‘𝑊)) = ∅)
5	4	raleqdv 3334	. . . 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 14581	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
10		1nn0 12569	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2947	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ≠ 0 ↔ ¬ (♯‘𝑊) = 0)
13		elnnne0 12567	. . . . . . . . . . . . . . . . 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 2954	. . . . . . . . . . . . . . 15 ⊢ (¬ (♯‘𝑊) = 1 → (♯‘𝑊) ≠ 1)
19	18	ad2antll 728	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑊) ∈ ℕ₀ ∧ (¬ (♯‘𝑊) = 0 ∧ ¬ (♯‘𝑊) = 1)) → (♯‘𝑊) ≠ 1)
20		nngt1ne1 12322	. . . . . . . . . . . . . . 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 13757	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0..^(♯‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 1 < (♯‘𝑊)))
24	11, 17, 22, 23	syl3anbrc 1343	. . . . . . . . . . . 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 13756	. . . . . . . . . . . . . . . 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 13716	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(♯‘𝑊)) → 1 ∈ ℤ)
40		cshweqrep 14869	. . . . . . . . . 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 592	. . . . . . . . 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 12568	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
45		fzossnn0 13747	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0..^(♯‘𝑊)) ⊆ ℕ₀)
46		ssralv 4077	. . . . . . . . 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 2747	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (♯‘𝑊))) = (𝑊‘0))
49		elfzoelz 13716	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → 𝑖 ∈ ℤ)
50		zre 12643	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
51		ax-1rid 11254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
52	50, 51	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
53	52	oveq2d 7464	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
54		zcn 12644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
55	54	addlidd 11491	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
56	53, 55	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
57	49, 56	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
58	57	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = (𝑖 mod (♯‘𝑊)))
59		zmodidfzoimp 13952	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → (𝑖 mod (♯‘𝑊)) = 𝑖)
60	58, 59	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(♯‘𝑊)) → ((0 + (𝑖 · 1)) mod (♯‘𝑊)) = 𝑖)
61	60	fveqeq2d 6928	. . . . . . . . . . 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 3086	. . . . . . . 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 2740	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
70		c0ex 11284	. . . . . . 7 ⊢ 0 ∈ V
71		fveqeq2 6929	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
72	70, 71	ralsn 4705	. . . . . 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 7456	. . . . . . 7 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = (0..^1))
75		fzo01 13798	. . . . . . 7 ⊢ (0..^1) = {0}
76	74, 75	eqtrdi 2796	. . . . . 6 ⊢ ((♯‘𝑊) = 1 → (0..^(♯‘𝑊)) = {0})
77	76	raleqdv 3334	. . . . 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 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ⊆ wss 3976 ∅c0 4352 {csn 4648 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ..^cfzo 13711 mod cmo 13920 ♯chash 14379 Word cword 14562 cyclShift ccsh 14836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-hash 14380 df-word 14563 df-concat 14619 df-substr 14689 df-pfx 14719 df-csh 14837

This theorem is referenced by: cshw1repsw 14871

W3C validator