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Theorem cshwrepswhash1 17032
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 12475 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2 repsdf2 14724 . . . . . . . 8 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)))
31, 2sylan2 593 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)))
4 simp1 1136 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 𝑊 ∈ Word 𝑉)
54adantl 482 . . . . . . . . 9 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → 𝑊 ∈ Word 𝑉)
6 eleq1 2821 . . . . . . . . . . . . . . . 16 (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
76eqcoms 2740 . . . . . . . . . . . . . . 15 ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
8 lbfzo0 13668 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
98biimpri 227 . . . . . . . . . . . . . . 15 ((♯‘𝑊) ∈ ℕ → 0 ∈ (0..^(♯‘𝑊)))
107, 9syl6bi 252 . . . . . . . . . . . . . 14 ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ → 0 ∈ (0..^(♯‘𝑊))))
11103ad2ant2 1134 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((𝐴𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 0 ∈ (0..^(♯‘𝑊))))
1413imp 407 . . . . . . . . . 10 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → 0 ∈ (0..^(♯‘𝑊)))
15 cshw0 14740 . . . . . . . . . . 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 7413 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑊 cyclShift 𝑛) = (𝑊 cyclShift 0))
1817eqeq1d 2734 . . . . . . . . . . 11 (𝑛 = 0 → ((𝑊 cyclShift 𝑛) = 𝑊 ↔ (𝑊 cyclShift 0) = 𝑊))
1918rspcev 3612 . . . . . . . . . 10 ((0 ∈ (0..^(♯‘𝑊)) ∧ (𝑊 cyclShift 0) = 𝑊) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
2014, 16, 19syl2anc 584 . . . . . . . . 9 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
21 eqeq2 2744 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑊))
2221rexbidv 3178 . . . . . . . . . 10 (𝑤 = 𝑊 → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊))
2322rspcev 3612 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
245, 20, 23syl2anc 584 . . . . . . . 8 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
2524ex 413 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
263, 25sylbid 239 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
27263impia 1117 . . . . 5 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
28 repsw 14721 . . . . . . . 8 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
291, 28sylan2 593 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
30293adant3 1132 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
31 simpll3 1214 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑊 = (𝐴 repeatS 𝑁))
3231oveq1d 7420 . . . . . . . . . . 11 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) = ((𝐴 repeatS 𝑁) cyclShift 𝑛))
33 simp1 1136 . . . . . . . . . . . . 13 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝐴𝑉)
3433ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝐴𝑉)
3513ad2ant2 1134 . . . . . . . . . . . . 13 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑁 ∈ ℕ0)
3635ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑁 ∈ ℕ0)
37 elfzoelz 13628 . . . . . . . . . . . . 13 (𝑛 ∈ (0..^(♯‘𝑊)) → 𝑛 ∈ ℤ)
3837adantl 482 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑛 ∈ ℤ)
39 repswcshw 14758 . . . . . . . . . . . 12 ((𝐴𝑉𝑁 ∈ ℕ0𝑛 ∈ ℤ) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4034, 36, 38, 39syl3anc 1371 . . . . . . . . . . 11 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4132, 40eqtrd 2772 . . . . . . . . . 10 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4241eqeq1d 2734 . . . . . . . . 9 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
4342biimpd 228 . . . . . . . 8 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
4443rexlimdva 3155 . . . . . . 7 (((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
4544ralrimiva 3146 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
46 eqeq1 2736 . . . . . . . . 9 (𝑤 = (𝐴 repeatS 𝑁) → (𝑤 = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
4746imbi2d 340 . . . . . . . 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 3177 . . . . . . 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 3612 . . . . . 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 584 . . . . 5 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢𝑤 = 𝑢))
51 eqeq2 2744 . . . . . . 7 (𝑤 = 𝑢 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑢))
5251rexbidv 3178 . . . . . 6 (𝑤 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢))
5352reu7 3727 . . . . 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 583 . . . 4 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
55 reusn 4730 . . . 4 (∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
5654, 55sylib 217 . . 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 2737 . . . 4 (𝑀 = {𝑟} ↔ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
5958exbii 1850 . . 3 (∃𝑟 𝑀 = {𝑟} ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
6056, 59sylibr 233 . 2 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑟 𝑀 = {𝑟})
6157cshwsex 17030 . . . . . 6 (𝑊 ∈ Word 𝑉𝑀 ∈ V)
62613ad2ant1 1133 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 𝑀 ∈ V)
633, 62syl6bi 252 . . . 4 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → 𝑀 ∈ V))
64633impia 1117 . . 3 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑀 ∈ V)
65 hash1snb 14375 . . 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 256 1 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (♯‘𝑀) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3061  wrex 3070  ∃!wreu 3374  {crab 3432  Vcvv 3474  {csn 4627  cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107  cn 12208  0cn0 12468  cz 12554  ..^cfzo 13623  chash 14286  Word cword 14460   repeatS creps 14714   cyclShift ccsh 14734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-hash 14287  df-word 14461  df-concat 14517  df-substr 14587  df-pfx 14617  df-reps 14715  df-csh 14735
This theorem is referenced by:  cshwshash  17034
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