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Theorem cshwrepswhash1 15589
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 11142 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2 repsdf2 13318 . . . . . . . 8 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)))
31, 2sylan2 489 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)))
4 simp1 1053 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 𝑊 ∈ Word 𝑉)
54adantl 480 . . . . . . . . 9 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → 𝑊 ∈ Word 𝑉)
6 eleq1 2671 . . . . . . . . . . . . . . . 16 (𝑁 = (#‘𝑊) → (𝑁 ∈ ℕ ↔ (#‘𝑊) ∈ ℕ))
76eqcoms 2613 . . . . . . . . . . . . . . 15 ((#‘𝑊) = 𝑁 → (𝑁 ∈ ℕ ↔ (#‘𝑊) ∈ ℕ))
8 lbfzo0 12326 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(#‘𝑊)) ↔ (#‘𝑊) ∈ ℕ)
98biimpri 216 . . . . . . . . . . . . . . 15 ((#‘𝑊) ∈ ℕ → 0 ∈ (0..^(#‘𝑊)))
107, 9syl6bi 241 . . . . . . . . . . . . . 14 ((#‘𝑊) = 𝑁 → (𝑁 ∈ ℕ → 0 ∈ (0..^(#‘𝑊))))
11103ad2ant2 1075 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝐴𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 0 ∈ (0..^(#‘𝑊))))
1413imp 443 . . . . . . . . . 10 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → 0 ∈ (0..^(#‘𝑊)))
15 cshw0 13333 . . . . . . . . . . 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 6531 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑊 cyclShift 𝑛) = (𝑊 cyclShift 0))
1817eqeq1d 2607 . . . . . . . . . . 11 (𝑛 = 0 → ((𝑊 cyclShift 𝑛) = 𝑊 ↔ (𝑊 cyclShift 0) = 𝑊))
1918rspcev 3277 . . . . . . . . . 10 ((0 ∈ (0..^(#‘𝑊)) ∧ (𝑊 cyclShift 0) = 𝑊) → ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
2014, 16, 19syl2anc 690 . . . . . . . . 9 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
21 eqeq2 2616 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑊))
2221rexbidv 3029 . . . . . . . . . 10 (𝑤 = 𝑊 → (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊))
2322rspcev 3277 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
245, 20, 23syl2anc 690 . . . . . . . 8 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
2524ex 448 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
263, 25sylbid 228 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
27263impia 1252 . . . . 5 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
28 repsw 13315 . . . . . . . 8 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
291, 28sylan2 489 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
30293adant3 1073 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
31 simpll3 1094 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → 𝑊 = (𝐴 repeatS 𝑁))
3231oveq1d 6538 . . . . . . . . . . 11 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑛) = ((𝐴 repeatS 𝑁) cyclShift 𝑛))
33 simp1 1053 . . . . . . . . . . . . 13 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝐴𝑉)
3433ad2antrr 757 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → 𝐴𝑉)
3513ad2ant2 1075 . . . . . . . . . . . . 13 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑁 ∈ ℕ0)
3635ad2antrr 757 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → 𝑁 ∈ ℕ0)
37 elfzoelz 12290 . . . . . . . . . . . . 13 (𝑛 ∈ (0..^(#‘𝑊)) → 𝑛 ∈ ℤ)
3837adantl 480 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → 𝑛 ∈ ℤ)
39 repswcshw 13351 . . . . . . . . . . . 12 ((𝐴𝑉𝑁 ∈ ℕ0𝑛 ∈ ℤ) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4034, 36, 38, 39syl3anc 1317 . . . . . . . . . . 11 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4132, 40eqtrd 2639 . . . . . . . . . 10 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4241eqeq1d 2607 . . . . . . . . 9 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
4342biimpd 217 . . . . . . . 8 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
4443rexlimdva 3008 . . . . . . 7 (((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) → (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
4544ralrimiva 2944 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
46 eqeq1 2609 . . . . . . . . 9 (𝑤 = (𝐴 repeatS 𝑁) → (𝑤 = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
4746imbi2d 328 . . . . . . . 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 2964 . . . . . . 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 3277 . . . . . 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 690 . . . . 5 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢𝑤 = 𝑢))
51 eqeq2 2616 . . . . . . 7 (𝑤 = 𝑢 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑢))
5251rexbidv 3029 . . . . . 6 (𝑤 = 𝑢 → (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢))
5352reu7 3363 . . . . 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 694 . . . 4 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
55 reusn 4201 . . . 4 (∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
5654, 55sylib 206 . . 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 2610 . . . 4 (𝑀 = {𝑟} ↔ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
5958exbii 1762 . . 3 (∃𝑟 𝑀 = {𝑟} ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
6056, 59sylibr 222 . 2 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑟 𝑀 = {𝑟})
6157cshwsex 15587 . . . . . 6 (𝑊 ∈ Word 𝑉𝑀 ∈ V)
62613ad2ant1 1074 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 𝑀 ∈ V)
633, 62syl6bi 241 . . . 4 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → 𝑀 ∈ V))
64633impia 1252 . . 3 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑀 ∈ V)
65 hash1snb 13016 . . 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 245 1 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (#‘𝑀) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1975  wral 2891  wrex 2892  ∃!wreu 2893  {crab 2895  Vcvv 3168  {csn 4120  cfv 5786  (class class class)co 6523  0cc0 9788  1c1 9789  cn 10863  0cn0 11135  cz 11206  ..^cfzo 12285  #chash 12930  Word cword 13088   repeatS creps 13095   cyclShift ccsh 13327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-fl 12406  df-mod 12482  df-hash 12931  df-word 13096  df-concat 13098  df-substr 13100  df-reps 13103  df-csh 13328
This theorem is referenced by:  cshwshash  15591
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