Theorem cshwrepswhash1 17137

Description: The size of the set of (different!) words resulting by cyclically shifting a nonempty "repeated symbol word" is 1. (Contributed by AV, 18-May-2018.) (Revised by AV, 8-Nov-2018.)

Hypothesis

Ref	Expression
cshwrepswhash1.m	⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}

Assertion

Ref	Expression
cshwrepswhash1	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (♯‘𝑀) = 1)

Distinct variable groups:   𝑛,𝑉,𝑤   𝑛,𝑊,𝑤   𝐴,𝑛,𝑤   𝑛,𝑁,𝑤

Allowed substitution hints:   𝑀(𝑤,𝑛)

Proof of Theorem cshwrepswhash1

Dummy variables 𝑖 𝑢 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 12531	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2		repsdf2 14813	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)))
3	1, 2	sylan2 593	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)))
4		simp1 1135	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → 𝑊 ∈ Word 𝑉)
5	4	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)) → 𝑊 ∈ Word 𝑉)
6		eleq1 2827	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
7	6	eqcoms 2743	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
8		lbfzo0 13736	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
9	8	biimpri 228	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) ∈ ℕ → 0 ∈ (0..^(♯‘𝑊)))
10	7, 9	biimtrdi 253	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ → 0 ∈ (0..^(♯‘𝑊))))
11	10	3ad2ant2 1133	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → (𝑁 ∈ ℕ → 0 ∈ (0..^(♯‘𝑊))))
12	11	com12 32	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → 0 ∈ (0..^(♯‘𝑊))))
13	12	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → 0 ∈ (0..^(♯‘𝑊))))
14	13	imp 406	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)) → 0 ∈ (0..^(♯‘𝑊)))
15		cshw0 14829	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)
16	5, 15	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)) → (𝑊 cyclShift 0) = 𝑊)
17		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑊 cyclShift 𝑛) = (𝑊 cyclShift 0))
18	17	eqeq1d 2737	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → ((𝑊 cyclShift 𝑛) = 𝑊 ↔ (𝑊 cyclShift 0) = 𝑊))
19	18	rspcev 3622	. . . . . . . . . 10 ⊢ ((0 ∈ (0..^(♯‘𝑊)) ∧ (𝑊 cyclShift 0) = 𝑊) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
20	14, 16, 19	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
21		eqeq2 2747	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑊))
22	21	rexbidv 3177	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊))
23	22	rspcev 3622	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊) → ∃𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
24	5, 20, 23	syl2anc 584	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)) → ∃𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
25	24	ex 412	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → ∃𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
26	3, 25	sylbid 240	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → ∃𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
27	26	3impia 1116	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
28		repsw 14810	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
29	1, 28	sylan2 593	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
30	29	3adant3 1131	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
31		simpll3 1213	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑊 = (𝐴 repeatS 𝑁))
32	31	oveq1d 7446	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) = ((𝐴 repeatS 𝑁) cyclShift 𝑛))
33		simp1 1135	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝐴 ∈ 𝑉)
34	33	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝐴 ∈ 𝑉)
35	1	3ad2ant2 1133	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑁 ∈ ℕ0)
36	35	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑁 ∈ ℕ0)
37		elfzoelz 13696	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0..^(♯‘𝑊)) → 𝑛 ∈ ℤ)
38	37	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑛 ∈ ℤ)
39		repswcshw 14847	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
40	34, 36, 38, 39	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
41	32, 40	eqtrd 2775	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) = (𝐴 repeatS 𝑁))
42	41	eqeq1d 2737	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
43	42	biimpd 229	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
44	43	rexlimdva 3153	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
45	44	ralrimiva 3144	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
46		eqeq1 2739	. . . . . . . . 9 ⊢ (𝑤 = (𝐴 repeatS 𝑁) → (𝑤 = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
47	46	imbi2d 340	. . . . . . . 8 ⊢ (𝑤 = (𝐴 repeatS 𝑁) → ((∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → 𝑤 = 𝑢) ↔ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢)))
48	47	ralbidv 3176	. . . . . . 7 ⊢ (𝑤 = (𝐴 repeatS 𝑁) → (∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → 𝑤 = 𝑢) ↔ ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢)))
49	48	rspcev 3622	. . . . . 6 ⊢ (((𝐴 repeatS 𝑁) ∈ Word 𝑉 ∧ ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢)) → ∃𝑤 ∈ Word 𝑉∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → 𝑤 = 𝑢))
50	30, 45, 49	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → 𝑤 = 𝑢))
51		eqeq2 2747	. . . . . . 7 ⊢ (𝑤 = 𝑢 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑢))
52	51	rexbidv 3177	. . . . . 6 ⊢ (𝑤 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢))
53	52	reu7 3741	. . . . 5 ⊢ (∃!𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ (∃𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ∧ ∃𝑤 ∈ Word 𝑉∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → 𝑤 = 𝑢)))
54	27, 50, 53	sylanbrc 583	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃!𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
55		reusn 4732	. . . 4 ⊢ (∃!𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
56	54, 55	sylib 218	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
57		cshwrepswhash1.m	. . . . 5 ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
58	57	eqeq1i 2740	. . . 4 ⊢ (𝑀 = {𝑟} ↔ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
59	58	exbii 1845	. . 3 ⊢ (∃𝑟 𝑀 = {𝑟} ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
60	56, 59	sylibr 234	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑟 𝑀 = {𝑟})
61	57	cshwsex 17135	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 ∈ V)
62	61	3ad2ant1 1132	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → 𝑀 ∈ V)
63	3, 62	biimtrdi 253	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → 𝑀 ∈ V))
64	63	3impia 1116	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑀 ∈ V)
65		hash1snb 14455	. . 3 ⊢ (𝑀 ∈ V → ((♯‘𝑀) = 1 ↔ ∃𝑟 𝑀 = {𝑟}))
66	64, 65	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ((♯‘𝑀) = 1 ↔ ∃𝑟 𝑀 = {𝑟}))
67	60, 66	mpbird 257	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (♯‘𝑀) = 1)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  ∃!wreu 3376  {crab 3433  Vcvv 3478  {csn 4631  ‘cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  ℕcn 12264  ℕ₀cn0 12524  ℤcz 12611  ..^cfzo 13691  ♯chash 14366  Word cword 14549   repeatS creps 14803   cyclShift ccsh 14823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-hash 14367  df-word 14550  df-concat 14606  df-substr 14676  df-pfx 14706  df-reps 14804  df-csh 14824

This theorem is referenced by:  cshwshash  17139