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Theorem cshwrepswhash1 16431
 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 11895 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
2 repsdf2 14134 . . . . . . . 8 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)))
31, 2sylan2 595 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)))
4 simp1 1133 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 𝑊 ∈ Word 𝑉)
54adantl 485 . . . . . . . . 9 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → 𝑊 ∈ Word 𝑉)
6 eleq1 2877 . . . . . . . . . . . . . . . 16 (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
76eqcoms 2806 . . . . . . . . . . . . . . 15 ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
8 lbfzo0 13075 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
98biimpri 231 . . . . . . . . . . . . . . 15 ((♯‘𝑊) ∈ ℕ → 0 ∈ (0..^(♯‘𝑊)))
107, 9syl6bi 256 . . . . . . . . . . . . . 14 ((♯‘𝑊) = 𝑁 → (𝑁 ∈ ℕ → 0 ∈ (0..^(♯‘𝑊))))
11103ad2ant2 1131 . . . . . . . . . . . . 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 485 . . . . . . . . . . 11 ((𝐴𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 0 ∈ (0..^(♯‘𝑊))))
1413imp 410 . . . . . . . . . 10 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → 0 ∈ (0..^(♯‘𝑊)))
15 cshw0 14150 . . . . . . . . . . 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 7144 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑊 cyclShift 𝑛) = (𝑊 cyclShift 0))
1817eqeq1d 2800 . . . . . . . . . . 11 (𝑛 = 0 → ((𝑊 cyclShift 𝑛) = 𝑊 ↔ (𝑊 cyclShift 0) = 𝑊))
1918rspcev 3571 . . . . . . . . . 10 ((0 ∈ (0..^(♯‘𝑊)) ∧ (𝑊 cyclShift 0) = 𝑊) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
2014, 16, 19syl2anc 587 . . . . . . . . 9 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊)
21 eqeq2 2810 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑊))
2221rexbidv 3256 . . . . . . . . . 10 (𝑤 = 𝑊 → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊))
2322rspcev 3571 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑊) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
245, 20, 23syl2anc 587 . . . . . . . 8 (((𝐴𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴)) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
2524ex 416 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
263, 25sylbid 243 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
27263impia 1114 . . . . 5 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
28 repsw 14131 . . . . . . . 8 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
291, 28sylan2 595 . . . . . . 7 ((𝐴𝑉𝑁 ∈ ℕ) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
30293adant3 1129 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (𝐴 repeatS 𝑁) ∈ Word 𝑉)
31 simpll3 1211 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑊 = (𝐴 repeatS 𝑁))
3231oveq1d 7151 . . . . . . . . . . 11 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) = ((𝐴 repeatS 𝑁) cyclShift 𝑛))
33 simp1 1133 . . . . . . . . . . . . 13 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝐴𝑉)
3433ad2antrr 725 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝐴𝑉)
3513ad2ant2 1131 . . . . . . . . . . . . 13 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑁 ∈ ℕ0)
3635ad2antrr 725 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑁 ∈ ℕ0)
37 elfzoelz 13036 . . . . . . . . . . . . 13 (𝑛 ∈ (0..^(♯‘𝑊)) → 𝑛 ∈ ℤ)
3837adantl 485 . . . . . . . . . . . 12 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → 𝑛 ∈ ℤ)
39 repswcshw 14168 . . . . . . . . . . . 12 ((𝐴𝑉𝑁 ∈ ℕ0𝑛 ∈ ℤ) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4034, 36, 38, 39syl3anc 1368 . . . . . . . . . . 11 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝐴 repeatS 𝑁) cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4132, 40eqtrd 2833 . . . . . . . . . 10 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) = (𝐴 repeatS 𝑁))
4241eqeq1d 2800 . . . . . . . . 9 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
4342biimpd 232 . . . . . . . 8 ((((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
4443rexlimdva 3243 . . . . . . 7 (((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) ∧ 𝑢 ∈ Word 𝑉) → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
4544ralrimiva 3149 . . . . . 6 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∀𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢 → (𝐴 repeatS 𝑁) = 𝑢))
46 eqeq1 2802 . . . . . . . . 9 (𝑤 = (𝐴 repeatS 𝑁) → (𝑤 = 𝑢 ↔ (𝐴 repeatS 𝑁) = 𝑢))
4746imbi2d 344 . . . . . . . 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 3162 . . . . . . 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 3571 . . . . . 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 587 . . . . 5 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑤 ∈ Word 𝑉𝑢 ∈ Word 𝑉(∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢𝑤 = 𝑢))
51 eqeq2 2810 . . . . . . 7 (𝑤 = 𝑢 → ((𝑊 cyclShift 𝑛) = 𝑤 ↔ (𝑊 cyclShift 𝑛) = 𝑢))
5251rexbidv 3256 . . . . . 6 (𝑤 = 𝑢 → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑢))
5352reu7 3671 . . . . 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 586 . . . 4 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
55 reusn 4623 . . . 4 (∃!𝑤 ∈ Word 𝑉𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
5654, 55sylib 221 . . 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 2803 . . . 4 (𝑀 = {𝑟} ↔ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
5958exbii 1849 . . 3 (∃𝑟 𝑀 = {𝑟} ↔ ∃𝑟{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑟})
6056, 59sylibr 237 . 2 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → ∃𝑟 𝑀 = {𝑟})
6157cshwsex 16429 . . . . . 6 (𝑊 ∈ Word 𝑉𝑀 ∈ V)
62613ad2ant1 1130 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝐴) → 𝑀 ∈ V)
633, 62syl6bi 256 . . . 4 ((𝐴𝑉𝑁 ∈ ℕ) → (𝑊 = (𝐴 repeatS 𝑁) → 𝑀 ∈ V))
64633impia 1114 . . 3 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → 𝑀 ∈ V)
65 hash1snb 13779 . . 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 260 1 ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (♯‘𝑀) = 1)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  {crab 3110  Vcvv 3441  {csn 4525  ‘cfv 6325  (class class class)co 7136  0cc0 10529  1c1 10530  ℕcn 11628  ℕ0cn0 11888  ℤcz 11972  ..^cfzo 13031  ♯chash 13689  Word cword 13860   repeatS creps 14124   cyclShift ccsh 14144 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-sup 8893  df-inf 8894  df-dju 9317  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-rp 12381  df-fz 12889  df-fzo 13032  df-fl 13160  df-mod 13236  df-hash 13690  df-word 13861  df-concat 13917  df-substr 13997  df-pfx 14027  df-reps 14125  df-csh 14145 This theorem is referenced by:  cshwshash  16433
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