cshw1 - Metamath Proof Explorer

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Theorem cshw1 13419

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 4028	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)
2		oveq2 6557	. . . . . 6 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = (0..^0))
3		fzo0 12361	. . . . . 6 ⊢ (0..^0) = ∅
4	2, 3	syl6eq 2660	. . . . 5 ⊢ ((#‘𝑊) = 0 → (0..^(#‘𝑊)) = ∅)
5	4	raleqdv 3121	. . . 4 ⊢ ((#‘𝑊) = 0 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑊‘0)))
6	1, 5	mpbiri 247	. . 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 790	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → 𝑊 ∈ Word 𝑉)
9		lencl 13179	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
10		1nn0 11185	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 ∈ ℕ₀)
12		df-ne 2782	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 0 ↔ ¬ (#‘𝑊) = 0)
13		elnnne0 11183	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℕ₀ ∧ (#‘𝑊) ≠ 0))
14	13	simplbi2com 655	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 0 → ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℕ))
15	12, 14	sylbir 224	. . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ≠ 1 ↔ ¬ (#‘𝑊) = 1)
19	18	biimpri 217	. . . . . . . . . . . . . . 15 ⊢ (¬ (#‘𝑊) = 1 → (#‘𝑊) ≠ 1)
20	19	ad2antll 761	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → (#‘𝑊) ≠ 1)
21		nngt1ne1 10924	. . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1)) → 1 < (#‘𝑊))
24		elfzo0 12376	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0..^(#‘𝑊)) ↔ (1 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ ∧ 1 < (#‘𝑊)))
25	11, 17, 23, 24	syl3anbrc 1239	. . . . . . . . . . . 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 792	. . . . . . . 8 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → (𝑊 cyclShift 1) = 𝑊)
31		lbfzo0 12375	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
32	31	biimpri 217	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) ∈ ℕ → 0 ∈ (0..^(#‘𝑊)))
33	13, 32	sylbir 224	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ (#‘𝑊) ≠ 0) → 0 ∈ (0..^(#‘𝑊)))
34	33	ex 449	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑊) ∈ ℕ₀ → ((#‘𝑊) ≠ 0 → 0 ∈ (0..^(#‘𝑊))))
35	12, 34	syl5bir 232	. . . . . . . . . . . . 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 12339	. . . . . . . . . 10 ⊢ (1 ∈ (0..^(#‘𝑊)) → 1 ∈ ℤ)
42		cshweqrep 13418	. . . . . . . . . 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 1319	. . . . . . 7 ⊢ (((¬ (#‘𝑊) = 0 ∧ ¬ (#‘𝑊) = 1) ∧ (𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊)) → ∀𝑖 ∈ ℕ₀ (𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))))
46		0nn0 11184	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
47		fzossnn0 12368	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (0..^(#‘𝑊)) ⊆ ℕ₀)
48		ssralv 3629	. . . . . . . . 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 2617	. . . . . . . . . 10 ⊢ ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) ↔ (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0))
51		elfzoelz 12339	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → 𝑖 ∈ ℤ)
52		zre 11258	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
53		ax-1rid 9885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℝ → (𝑖 · 1) = 𝑖)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → (𝑖 · 1) = 𝑖)
55	54	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = (0 + 𝑖))
56		zcn 11259	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ)
57	56	addid2d 10116	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → (0 + 𝑖) = 𝑖)
58	55, 57	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0 + (𝑖 · 1)) = 𝑖)
59	51, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (0 + (𝑖 · 1)) = 𝑖)
60	59	oveq1d 6564	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((0 + (𝑖 · 1)) mod (#‘𝑊)) = (𝑖 mod (#‘𝑊)))
61		zmodidfzoimp 12562	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (𝑖 mod (#‘𝑊)) = 𝑖)
62	60, 61	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((0 + (𝑖 · 1)) mod (#‘𝑊)) = 𝑖)
63	62	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘𝑖))
64	63	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0) ↔ (𝑊‘𝑖) = (𝑊‘0)))
65	64	biimpd 218	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) = (𝑊‘0) → (𝑊‘𝑖) = (𝑊‘0)))
66	50, 65	syl5bi 231	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^(#‘𝑊)) → ((𝑊‘0) = (𝑊‘((0 + (𝑖 · 1)) mod (#‘𝑊))) → (𝑊‘𝑖) = (𝑊‘0)))
67	66	ralimia 2934	. . . . . . . 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 2610	. . . . . 6 ⊢ (𝑊‘0) = (𝑊‘0)
73		c0ex 9913	. . . . . . 7 ⊢ 0 ∈ V
74		fveq2 6103	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑊‘𝑖) = (𝑊‘0))
75	74	eqeq1d 2612	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0)))
76	73, 75	ralsn 4169	. . . . . 6 ⊢ (∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0) ↔ (𝑊‘0) = (𝑊‘0))
77	72, 76	mpbir 220	. . . . 5 ⊢ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)
78		oveq2 6557	. . . . . . 7 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = (0..^1))
79		fzo01 12417	. . . . . . 7 ⊢ (0..^1) = {0}
80	78, 79	syl6eq 2660	. . . . . 6 ⊢ ((#‘𝑊) = 1 → (0..^(#‘𝑊)) = {0})
81	80	raleqdv 3121	. . . . 5 ⊢ ((#‘𝑊) = 1 → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0) ↔ ∀𝑖 ∈ {0} (𝑊‘𝑖) = (𝑊‘0)))
82	77, 81	mpbiri 247	. . . 4 ⊢ ((#‘𝑊) = 1 → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊‘𝑖) = (𝑊‘0))
83	71, 82	pm2.61d2 171	. . 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 175	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ..^cfzo 12334 mod cmo 12530 #chash 12979 Word cword 13146 cyclShift ccsh 13385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-csh 13386

This theorem is referenced by: cshw1repsw 13420

W3C validator