cshw1 - Metamath Proof Explorer

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Theorem cshw1 13614

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 4109	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 6698	. . . . . 6 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = (0..^0))
3		fzo0 12531	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	syl6eq 2701	. . . . 5 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = ∅)
5	4	raleqdv 3174	. . . 4 ⊢ ((#‘𝑊) = 0 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)))
6	1, 5	mpbiri 248	. . 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 809	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 𝑊 ∈ Word 𝑉)
9		lencl 13356	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
10		1nn0 11346	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2824	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 0 ↔ ¬ (#‘𝑊) = 0)
13		elnnne0 11344	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℕ₀ ∧ (#‘𝑊) ≠ 0))
14	13	simplbi2com 656	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 0 → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
15	12, 14	sylbir 225	. . . . . . . . . . . . . . 15 ⊢ (¬ (#‘𝑊) = 0 → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
16	15	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
17	16	impcom 445	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (#‘𝑊) ∈ ℕ)
18		df-ne 2824	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 1 ↔ ¬ (#‘𝑊) = 1)
19	18	biimpri 218	. . . . . . . . . . . . . . 15 ⊢ (¬ (#‘𝑊) = 1 → (#‘𝑊) ≠ 1)
20	19	ad2antll 765	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (#‘𝑊) ≠ 1)
21		nngt1ne1 11085	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑊) ∈ ℕ → (1 < (#‘𝑊) ↔ (#‘𝑊) ≠ 1))
22	17, 21	syl 17	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (1 < (#‘𝑊) ↔ (#‘𝑊) ≠ 1))
23	20, 22	mpbird 247	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 < (#‘𝑊))
24		elfzo0 12548	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0..^(#‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 1 < (#‘𝑊)))
25	11, 17, 23, 24	syl3anbrc 1265	. . . . . . . . . . . 12 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 ∈ (0..^(#‘𝑊)))
26	25	ex 449	. . . . . . . . . . 11 ⊢ ((#‘𝑊) ∈ ℕ₀ → ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → 1 ∈ (0..^(#‘𝑊))))
27	9, 26	syl 17	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → 1 ∈ (0..^(#‘𝑊))))
28	27	adantr 480	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → 1 ∈ (0..^(#‘𝑊))))
29	28	impcom 445	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 1 ∈ (0..^(#‘𝑊)))
30		simprr 811	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (𝑊 cyclShift 1) = 𝑊)
31		lbfzo0 12547	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
32	31	biimpri 218	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ∈ ℕ → 0 ∈ (0..^(#‘𝑊)))
33	13, 32	sylbir 225	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (#‘𝑊) ≠ 0) → 0 ∈ (0..^(#‘𝑊)))
34	33	ex 449	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑊) ∈ ℕ₀ → ((#‘𝑊) ≠ 0 → 0 ∈ (0..^(#‘𝑊))))
35	12, 34	syl5bir 233	. . . . . . . . . . . . 13 ⊢ ((#‘𝑊) ∈ ℕ₀ → (¬ (#‘𝑊) = 0 → 0 ∈ (0..^(#‘𝑊))))
36	9, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → (¬ (#‘𝑊) = 0 → 0 ∈ (0..^(#‘𝑊))))
37	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → (¬ (#‘𝑊) = 0 → 0 ∈ (0..^(#‘𝑊))))
38	37	com12 32	. . . . . . . . . 10 ⊢ (¬ (#‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(#‘𝑊))))
39	38	adantr 480	. . . . . . . . 9 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(#‘𝑊))))
40	39	imp 444	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 0 ∈ (0..^(#‘𝑊)))
41		elfzoelz 12509	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(#‘𝑊)) → 1 ∈ ℤ)
42		cshweqrep 13613	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ ℤ) → (((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(#‘𝑊))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊)))))
43	41, 42	sylan2 490	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(#‘𝑊))) → (((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(#‘𝑊))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊)))))
44	43	imp 444	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 1 ∈ (0..^(#‘𝑊))) ∧ ((𝑊 cyclShift 1) = 𝑊 ∧ 0 ∈ (0..^(#‘𝑊)))) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))))
45	8, 29, 30, 40, 44	syl22anc 1367	. . . . . . 7 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))))
46		0nn0 11345	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
47		fzossnn0 12538	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0..^(#‘𝑊)) ⊆ ℕ₀)
48		ssralv 3699	. . . . . . . . 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)))))
49	46, 47, 48	mp2b 10	. . . . . . . 8 ⊢ (∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))))
50		eqcom 2658	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0))
51		elfzoelz 12509	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → 𝑖 ∈ ℤ)
52		zre 11419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
53		ax-1rid 10044	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
55	54	oveq2d 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
56		zcn 11420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
57	56	addid2d 10275	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
58	55, 57	eqtrd 2685	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
59	51, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
60	59	oveq1d 6705	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((0 + (𝑖 · 1)) mod (#‘𝑊)) = (𝑖 mod (#‘𝑊)))
61		zmodidfzoimp 12740	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (𝑖 mod (#‘𝑊)) = 𝑖)
62	60, 61	eqtrd 2685	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((0 + (𝑖 · 1)) mod (#‘𝑊)) = 𝑖)
63	62	fveq2d 6233	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘𝑖))
64	63	eqeq1d 2653	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0) ↔ (𝑊‘𝑖) = (𝑊‘0)))
65	64	biimpd 219	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0) → (𝑊‘𝑖) = (𝑊‘0)))
66	50, 65	syl5bi 232	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
67	66	ralimia 2979	. . . . . . . 8 ⊢ (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
68	49, 67	syl 17	. . . . . . 7 ⊢ (∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
69	45, 68	syl 17	. . . . . 6 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
70	69	ex 449	. . . . 5 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
71	70	impancom 455	. . . 4 ⊢ ((¬ (#‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (¬ (#‘𝑊) = 1 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
72		eqid 2651	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
73		c0ex 10072	. . . . . . 7 ⊢ 0 ∈ V
74		fveq2 6229	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑊‘𝑖) = (𝑊‘0))
75	74	eqeq1d 2653	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
76	73, 75	ralsn 4254	. . . . . 6 ⊢ (∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0))
77	72, 76	mpbir 221	. . . . 5 ⊢ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)
78		oveq2 6698	. . . . . . 7 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = (0..^1))
79		fzo01 12590	. . . . . . 7 ⊢ (0..^1) = {0}
80	78, 79	syl6eq 2701	. . . . . 6 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = {0})
81	80	raleqdv 3174	. . . . 5 ⊢ ((#‘𝑊) = 1 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)))
82	77, 81	mpbiri 248	. . . 4 ⊢ ((#‘𝑊) = 1 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
83	71, 82	pm2.61d2 172	. . 3 ⊢ ((¬ (#‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
84	83	ex 449	. 2 ⊢ (¬ (#‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
85	7, 84	pm2.61i 176	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 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 ⊆ wss 3607 ∅c0 3948 {csn 4210 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 < clt 10112 ℕcn 11058 ℕ₀cn0 11330 ℤcz 11415 ..^cfzo 12504 mod cmo 12708 #chash 13157 Word cword 13323 cyclShift ccsh 13580

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-hash 13158 df-word 13331 df-concat 13333 df-substr 13335 df-csh 13581

This theorem is referenced by: cshw1repsw 13615

W3C validator