cshw1 - Metamath Proof Explorer

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Theorem cshw1 13367

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 4027	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 6534	. . . . . 6 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = (0..^0))
3		fzo0 12318	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	syl6eq 2659	. . . . 5 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = ∅)
5	4	raleqdv 3120	. . . 4 ⊢ ((#‘𝑊) = 0 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)))
6	1, 5	mpbiri 246	. . 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 789	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 𝑊 ∈ Word 𝑉)
9		lencl 13127	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
10		1nn0 11157	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2781	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 0 ↔ ¬ (#‘𝑊) = 0)
13		elnnne0 11155	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℕ₀ ∧ (#‘𝑊) ≠ 0))
14	13	simplbi2com 654	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 0 → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
15	12, 14	sylbir 223	. . . . . . . . . . . . . . 15 ⊢ (¬ (#‘𝑊) = 0 → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
16	15	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
17	16	impcom 444	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (#‘𝑊) ∈ ℕ)
18		df-ne 2781	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 1 ↔ ¬ (#‘𝑊) = 1)
19	18	biimpri 216	. . . . . . . . . . . . . . 15 ⊢ (¬ (#‘𝑊) = 1 → (#‘𝑊) ≠ 1)
20	19	ad2antll 760	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (#‘𝑊) ≠ 1)
21		nngt1ne1 10896	. . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 < (#‘𝑊))
24		elfzo0 12333	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0..^(#‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 1 < (#‘𝑊)))
25	11, 17, 23, 24	syl3anbrc 1238	. . . . . . . . . . . 12 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 ∈ (0..^(#‘𝑊)))
26	25	ex 448	. . . . . . . . . . 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 479	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → 1 ∈ (0..^(#‘𝑊))))
29	28	impcom 444	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 1 ∈ (0..^(#‘𝑊)))
30		simprr 791	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (𝑊 cyclShift 1) = 𝑊)
31		lbfzo0 12332	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
32	31	biimpri 216	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ∈ ℕ → 0 ∈ (0..^(#‘𝑊)))
33	13, 32	sylbir 223	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (#‘𝑊) ≠ 0) → 0 ∈ (0..^(#‘𝑊)))
34	33	ex 448	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑊) ∈ ℕ₀ → ((#‘𝑊) ≠ 0 → 0 ∈ (0..^(#‘𝑊))))
35	12, 34	syl5bir 231	. . . . . . . . . . . . 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 479	. . . . . . . . . . 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 479	. . . . . . . . 9 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 0 ∈ (0..^(#‘𝑊))))
40	39	imp 443	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 0 ∈ (0..^(#‘𝑊)))
41		elfzoelz 12296	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(#‘𝑊)) → 1 ∈ ℤ)
42		cshweqrep 13366	. . . . . . . . . 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 489	. . . . . . . . 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 443	. . . . . . . 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 1318	. . . . . . 7 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))))
46		0nn0 11156	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
47		fzossnn0 12325	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0..^(#‘𝑊)) ⊆ ℕ₀)
48		ssralv 3628	. . . . . . . . 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 2616	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0))
51		elfzoelz 12296	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → 𝑖 ∈ ℤ)
52		zre 11216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
53		ax-1rid 9862	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
55	54	oveq2d 6542	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
56		zcn 11217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
57	56	addid2d 10088	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
58	55, 57	eqtrd 2643	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
59	51, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
60	59	oveq1d 6541	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((0 + (𝑖 · 1)) mod (#‘𝑊)) = (𝑖 mod (#‘𝑊)))
61		zmodidfzoimp 12519	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (𝑖 mod (#‘𝑊)) = 𝑖)
62	60, 61	eqtrd 2643	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((0 + (𝑖 · 1)) mod (#‘𝑊)) = 𝑖)
63	62	fveq2d 6091	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘𝑖))
64	63	eqeq1d 2611	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0) ↔ (𝑊‘𝑖) = (𝑊‘0)))
65	64	biimpd 217	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0) → (𝑊‘𝑖) = (𝑊‘0)))
66	50, 65	syl5bi 230	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
67	66	ralimia 2933	. . . . . . . 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 448	. . . . 5 ⊢ ((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
71	70	impancom 454	. . . 4 ⊢ ((¬ (#‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (¬ (#‘𝑊) = 1 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
72		eqid 2609	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
73		c0ex 9890	. . . . . . 7 ⊢ 0 ∈ V
74		fveq2 6087	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑊‘𝑖) = (𝑊‘0))
75	74	eqeq1d 2611	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
76	73, 75	ralsn 4168	. . . . . 6 ⊢ (∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0))
77	72, 76	mpbir 219	. . . . 5 ⊢ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)
78		oveq2 6534	. . . . . . 7 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = (0..^1))
79		fzo01 12374	. . . . . . 7 ⊢ (0..^1) = {0}
80	78, 79	syl6eq 2659	. . . . . 6 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = {0})
81	80	raleqdv 3120	. . . . 5 ⊢ ((#‘𝑊) = 1 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)))
82	77, 81	mpbiri 246	. . . 4 ⊢ ((#‘𝑊) = 1 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
83	71, 82	pm2.61d2 170	. . 3 ⊢ ((¬ (#‘𝑊) = 0 ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
84	83	ex 448	. 2 ⊢ (¬ (#‘𝑊) = 0 → ((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0)))
85	7, 84	pm2.61i 174	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 194 ∧ wa 382 = wceq 1474 ∈ wcel 1976 ≠ wne 2779 ∀wral 2895 ⊆ wss 3539 ∅c0 3873 {csn 4124 class class class wbr 4577 ‘cfv 5789 (class class class)co 6526 ℝcr 9791 0cc0 9792 1c1 9793 + caddc 9795 · cmul 9797 < clt 9930 ℕcn 10869 ℕ₀cn0 11141 ℤcz 11212 ..^cfzo 12291 mod cmo 12487 #chash 12936 Word cword 13094 cyclShift ccsh 13333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4711 ax-pow 4763 ax-pr 4827 ax-un 6824 ax-cnex 9848 ax-resscn 9849 ax-1cn 9850 ax-icn 9851 ax-addcl 9852 ax-addrcl 9853 ax-mulcl 9854 ax-mulrcl 9855 ax-mulcom 9856 ax-addass 9857 ax-mulass 9858 ax-distr 9859 ax-i2m1 9860 ax-1ne0 9861 ax-1rid 9862 ax-rnegex 9863 ax-rrecex 9864 ax-cnre 9865 ax-pre-lttri 9866 ax-pre-lttrn 9867 ax-pre-ltadd 9868 ax-pre-mulgt0 9869 ax-pre-sup 9870

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-uni 4367 df-int 4405 df-iun 4451 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4938 df-id 4942 df-po 4948 df-so 4949 df-fr 4986 df-we 4988 df-xp 5033 df-rel 5034 df-cnv 5035 df-co 5036 df-dm 5037 df-rn 5038 df-res 5039 df-ima 5040 df-pred 5582 df-ord 5628 df-on 5629 df-lim 5630 df-suc 5631 df-iota 5753 df-fun 5791 df-fn 5792 df-f 5793 df-f1 5794 df-fo 5795 df-f1o 5796 df-fv 5797 df-riota 6488 df-ov 6529 df-oprab 6530 df-mpt2 6531 df-om 6935 df-1st 7036 df-2nd 7037 df-wrecs 7271 df-recs 7332 df-rdg 7370 df-1o 7424 df-oadd 7428 df-er 7606 df-en 7819 df-dom 7820 df-sdom 7821 df-fin 7822 df-sup 8208 df-inf 8209 df-card 8625 df-pnf 9932 df-mnf 9933 df-xr 9934 df-ltxr 9935 df-le 9936 df-sub 10119 df-neg 10120 df-div 10536 df-nn 10870 df-2 10928 df-n0 11142 df-z 11213 df-uz 11522 df-rp 11667 df-fz 12155 df-fzo 12292 df-fl 12412 df-mod 12488 df-hash 12937 df-word 13102 df-concat 13104 df-substr 13106 df-csh 13334

This theorem is referenced by: cshw1repsw 13368

W3C validator