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Theorem cshwrepswhash1 15856
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 ((𝐴𝑉𝑁 ∈ 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.
StepHypRef Expression
1 nnnn0 11337 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2 repsdf2 13571 . . . . . . . 8 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)))
31, 2sylan2 490 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)))
4 simp1 1081 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 𝑊 ∈ Word 𝑉)
54adantl 481 . . . . . . . . 9 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → 𝑊 ∈ Word 𝑉)
6 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑁 = (#‘𝑊) → (𝑁 ∈ ℕ ↔ (#‘𝑊) ∈ ℕ))
76eqcoms 2659 . . . . . . . . . . . . . . 15 ((#‘𝑊) = 𝑁 → (𝑁 ∈ ℕ ↔ (#‘𝑊) ∈ ℕ))
8 lbfzo0 12547 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
98biimpri 218 . . . . . . . . . . . . . . 15 ((#‘𝑊) ∈ ℕ → 0 ∈ (0..^(#‘𝑊)))
107, 9syl6bi 243 . . . . . . . . . . . . . 14 ((#‘𝑊) = 𝑁 → (𝑁 ∈ ℕ → 0 ∈ (0..^(#‘𝑊))))
11103ad2ant2 1103 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → (𝑁 ∈ ℕ → 0 ∈ (0..^(#‘𝑊))))
1211com12 32 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 0 ∈ (0..^(#‘𝑊))))
1312adantl 481 . . . . . . . . . . 11 ((𝐴𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 0 ∈ (0..^(#‘𝑊))))
1413imp 444 . . . . . . . . . 10 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → 0 ∈ (0..^(#‘𝑊)))
15 cshw0 13586 . . . . . . . . . . 11 (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)
165, 15syl 17 . . . . . . . . . 10 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → (𝑊 cyclShift 0) = 𝑊)
17 oveq2 6698 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑊 cyclShift 𝑛) = (𝑊 cyclShift 0))
1817eqeq1d 2653 . . . . . . . . . . 11 (𝑛 = 0 → ((𝑊 cyclShift 𝑛) = 𝑊 ↔ (𝑊 cyclShift 0) = 𝑊))
1918rspcev 3340 . . . . . . . . . 10 ((0 ∈ (0..^(#‘𝑊)) ∧ (𝑊 cyclShift 0) = 𝑊) → ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
2014, 16, 19syl2anc 694 . . . . . . . . 9 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
21 eqeq2 2662 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑊))
2221rexbidv 3081 . . . . . . . . . 10 (𝑤 = 𝑊 → (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊))
2322rspcev 3340 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
245, 20, 23syl2anc 694 . . . . . . . 8 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
2524ex 449 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
263, 25sylbid 230 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
27263impia 1280 . . . . 5 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
28 repsw 13568 . . . . . . . 8 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
291, 28sylan2 490 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
30293adant3 1101 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
31 simpll3 1122 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → 𝑊 = (𝐴 repeatS 𝑁))
3231oveq1d 6705 . . . . . . . . . . 11 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑛) = ((𝐴 repeatS 𝑁) cyclShift 𝑛))
33 simp1 1081 . . . . . . . . . . . . 13 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝐴𝑉)
3433ad2antrr 762 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → 𝐴𝑉)
3513ad2ant2 1103 . . . . . . . . . . . . 13 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑁 ∈ ℕ0)
3635ad2antrr 762 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → 𝑁 ∈ ℕ0)
37 elfzoelz 12509 . . . . . . . . . . . . 13 (𝑛 ∈ (0..^(#‘𝑊)) → 𝑛 ∈ ℤ)
3837adantl 481 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → 𝑛 ∈ ℤ)
39 repswcshw 13604 . . . . . . . . . . . 12 ((𝐴𝑉𝑁 ∈ ℕ0𝑛 ∈ ℤ) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4034, 36, 38, 39syl3anc 1366 . . . . . . . . . . 11 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4132, 40eqtrd 2685 . . . . . . . . . 10 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4241eqeq1d 2653 . . . . . . . . 9 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
4342biimpd 219 . . . . . . . 8 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
4443rexlimdva 3060 . . . . . . 7 (((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) → (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
4544ralrimiva 2995 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
46 eqeq1 2655 . . . . . . . . 9 (𝑤 = (𝐴 repeatS 𝑁) → (𝑤 = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
4746imbi2d 329 . . . . . . . 8 (𝑤 = (𝐴 repeatS 𝑁) → ((∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢𝑤 = 𝑢) ↔ (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢)))
4847ralbidv 3015 . . . . . . 7 (𝑤 = (𝐴 repeatS 𝑁) → (∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢𝑤 = 𝑢) ↔ ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢)))
4948rspcev 3340 . . . . . 6 (((𝐴 repeatS 𝑁) ∈ Word 𝑉 ∧ ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢)) → ∃𝑤 ∈ Word 𝑉𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢𝑤 = 𝑢))
5030, 45, 49syl2anc 694 . . . . 5 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢𝑤 = 𝑢))
51 eqeq2 2662 . . . . . . 7 (𝑤 = 𝑢 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑢))
5251rexbidv 3081 . . . . . 6 (𝑤 = 𝑢 → (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢))
5352reu7 3434 . . . . 5 (∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ (∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ∧ ∃𝑤 ∈ Word 𝑉𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢𝑤 = 𝑢)))
5427, 50, 53sylanbrc 699 . . . 4 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
55 reusn 4294 . . . 4 (∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
5654, 55sylib 208 . . 3 ((𝐴𝑉𝑁 ∈ 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}
5857eqeq1i 2656 . . . 4 (𝑀 = {𝑟} ↔ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
5958exbii 1814 . . 3 (∃𝑟 𝑀 = {𝑟} ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
6056, 59sylibr 224 . 2 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑟 𝑀 = {𝑟})
6157cshwsex 15854 . . . . . 6 (𝑊 ∈ Word 𝑉𝑀 ∈ V)
62613ad2ant1 1102 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 𝑀 ∈ V)
633, 62syl6bi 243 . . . 4 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → 𝑀 ∈ V))
64633impia 1280 . . 3 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑀 ∈ V)
65 hash1snb 13245 . . 3 (𝑀 ∈ V → ((#‘𝑀) = 1 ↔ ∃𝑟 𝑀 = {𝑟}))
6664, 65syl 17 . 2 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ((#‘𝑀) = 1 ↔ ∃𝑟 𝑀 = {𝑟}))
6760, 66mpbird 247 1 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (#‘𝑀) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  {crab 2945  Vcvv 3231  {csn 4210  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975  cn 11058  0cn0 11330  cz 11415  ..^cfzo 12504  #chash 13157  Word cword 13323   repeatS creps 13330   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-cda 9028  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-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-reps 13338  df-csh 13581
This theorem is referenced by:  cshwshash  15858
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