Theorem cshwrepswhash1 16269

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 11758	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2		repsdf2 13980	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)))
3	1, 2	sylan2 592	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)))
4		simp1 1129	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → 𝑊 ∈ Word 𝑉)
5	4	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)) → 𝑊 ∈ Word 𝑉)
6		eleq1 2872	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
7	6	eqcoms 2805	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
8		lbfzo0 12931	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
9	8	biimpri 229	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑊) ∈ ℕ → 0 ∈ (0..^(♯‘𝑊)))
10	7, 9	syl6bi 254	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ → 0 ∈ (0..^(♯‘𝑊))))
11	10	3ad2ant2 1127	. . . . . . . . . . . . 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 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → 0 ∈ (0..^(♯‘𝑊))))
14	13	imp 407	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴)) → 0 ∈ (0..^(♯‘𝑊)))
15		cshw0 13996	. . . . . . . . . . 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 7031	. . . . . . . . . . . 12 ⊢ (𝑛 = 0 → (𝑊 cyclShift 𝑛) = (𝑊 cyclShift 0))
18	17	eqeq1d 2799	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → ((𝑊 cyclShift 𝑛) = 𝑊 ↔ (𝑊 cyclShift 0) = 𝑊))
19	18	rspcev 3561	. . . . . . . . . 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 2808	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑊))
22	21	rexbidv 3262	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊))
23	22	rspcev 3561	. . . . . . . . 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 413	. . . . . . 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 241	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → ∃𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
27	26	3impia 1110	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
28		repsw 13977	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
29	1, 28	sylan2 592	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
30	29	3adant3 1125	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
31		simpll3 1207	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑊 = (𝐴 repeatS 𝑁))
32	31	oveq1d 7038	. . . . . . . . . . 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 1129	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝐴 ∈ 𝑉)
34	33	ad2antrr 722	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝐴 ∈ 𝑉)
35	1	3ad2ant2 1127	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑁 ∈ ℕ0)
36	35	ad2antrr 722	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑁 ∈ ℕ0)
37		elfzoelz 12892	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (0..^(♯‘𝑊)) → 𝑛 ∈ ℤ)
38	37	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑛 ∈ ℤ)
39		repswcshw 14014	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝑛 ∈ ℤ) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
40	34, 36, 38, 39	syl3anc 1364	. . . . . . . . . . 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 2833	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) = (𝐴 repeatS 𝑁))
42	41	eqeq1d 2799	. . . . . . . . 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 230	. . . . . . . 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 3249	. . . . . . 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 3151	. . . . . 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 2801	. . . . . . . . 9 ⊢ (𝑤 = (𝐴 repeatS 𝑁) → (𝑤 = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
47	46	imbi2d 342	. . . . . . . 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 3166	. . . . . . 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 3561	. . . . . 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 2808	. . . . . . 7 ⊢ (𝑤 = 𝑢 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑢))
52	51	rexbidv 3262	. . . . . 6 ⊢ (𝑤 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢))
53	52	reu7 3662	. . . . 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 4576	. . . 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 219	. . 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 2802	. . . 4 ⊢ (𝑀 = {𝑟} ↔ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
59	58	exbii 1833	. . 3 ⊢ (∃𝑟 𝑀 = {𝑟} ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
60	56, 59	sylibr 235	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑟 𝑀 = {𝑟})
61	57	cshwsex 16267	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 ∈ V)
62	61	3ad2ant1 1126	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝐴) → 𝑀 ∈ V)
63	3, 62	syl6bi 254	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → 𝑀 ∈ V))
64	63	3impia 1110	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑀 ∈ V)
65		hash1snb 13632	. . 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 258	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (♯‘𝑀) = 1)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525  ∃wex 1765   ∈ wcel 2083  ∀wral 3107  ∃wrex 3108  ∃!wreu 3109  {crab 3111  Vcvv 3440  {csn 4478  ‘cfv 6232  (class class class)co 7023  0cc0 10390  1c1 10391  ℕcn 11492  ℕ₀cn0 11751  ℤcz 11835  ..^cfzo 12887  ♯chash 13544  Word cword 13711   repeatS creps 13970   cyclShift ccsh 13990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-inf 8760  df-dju 9183  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-n0 11752  df-z 11836  df-uz 12098  df-rp 12244  df-fz 12747  df-fzo 12888  df-fl 13016  df-mod 13092  df-hash 13545  df-word 13712  df-concat 13773  df-substr 13843  df-pfx 13873  df-reps 13971  df-csh 13991

This theorem is referenced by:  cshwshash  16271