Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cycpmco2f1 Structured version   Visualization version   GIF version

Theorem cycpmco2f1 30766
 Description: The word U used in cycpmco2 30775 is injective, so it can represent a cycle and form a cyclic permutation (𝑀‘𝑈). (Contributed by Thierry Arnoux, 4-Jan-2024.)
Hypotheses
Ref Expression
cycpmco2.c 𝑀 = (toCyc‘𝐷)
cycpmco2.s 𝑆 = (SymGrp‘𝐷)
cycpmco2.d (𝜑𝐷𝑉)
cycpmco2.w (𝜑𝑊 ∈ dom 𝑀)
cycpmco2.i (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))
cycpmco2.j (𝜑𝐽 ∈ ran 𝑊)
cycpmco2.e 𝐸 = ((𝑊𝐽) + 1)
cycpmco2.1 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
Assertion
Ref Expression
cycpmco2f1 (𝜑𝑈:dom 𝑈1-1𝐷)

Proof of Theorem cycpmco2f1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 cycpmco2.d . . 3 (𝜑𝐷𝑉)
2 ssrab2 4055 . . . . . 6 {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷} ⊆ Word 𝐷
3 cycpmco2.w . . . . . . 7 (𝜑𝑊 ∈ dom 𝑀)
4 cycpmco2.c . . . . . . . . . 10 𝑀 = (toCyc‘𝐷)
5 cycpmco2.s . . . . . . . . . 10 𝑆 = (SymGrp‘𝐷)
6 eqid 2821 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
74, 5, 6tocycf 30759 . . . . . . . . 9 (𝐷𝑉𝑀:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶(Base‘𝑆))
81, 7syl 17 . . . . . . . 8 (𝜑𝑀:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶(Base‘𝑆))
98fdmd 6522 . . . . . . 7 (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
103, 9eleqtrd 2915 . . . . . 6 (𝜑𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
112, 10sseldi 3964 . . . . 5 (𝜑𝑊 ∈ Word 𝐷)
12 pfxcl 14038 . . . . 5 (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
1311, 12syl 17 . . . 4 (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
14 cycpmco2.i . . . . . 6 (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))
1514eldifad 3947 . . . . 5 (𝜑𝐼𝐷)
1615s1cld 13956 . . . 4 (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
17 ccatcl 13925 . . . 4 (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
1813, 16, 17syl2anc 586 . . 3 (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
19 swrdcl 14006 . . . 4 (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
2011, 19syl 17 . . 3 (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
21 id 22 . . . . . . . . 9 (𝑤 = 𝑊𝑤 = 𝑊)
22 dmeq 5771 . . . . . . . . 9 (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
23 eqidd 2822 . . . . . . . . 9 (𝑤 = 𝑊𝐷 = 𝐷)
2421, 22, 23f1eq123d 6607 . . . . . . . 8 (𝑤 = 𝑊 → (𝑤:dom 𝑤1-1𝐷𝑊:dom 𝑊1-1𝐷))
2524elrab 3679 . . . . . . 7 (𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷} ↔ (𝑊 ∈ Word 𝐷𝑊:dom 𝑊1-1𝐷))
2610, 25sylib 220 . . . . . 6 (𝜑 → (𝑊 ∈ Word 𝐷𝑊:dom 𝑊1-1𝐷))
2726simprd 498 . . . . 5 (𝜑𝑊:dom 𝑊1-1𝐷)
28 cycpmco2.e . . . . . 6 𝐸 = ((𝑊𝐽) + 1)
29 f1cnv 6637 . . . . . . . . . 10 (𝑊:dom 𝑊1-1𝐷𝑊:ran 𝑊1-1-onto→dom 𝑊)
30 f1of 6614 . . . . . . . . . 10 (𝑊:ran 𝑊1-1-onto→dom 𝑊𝑊:ran 𝑊⟶dom 𝑊)
3127, 29, 303syl 18 . . . . . . . . 9 (𝜑𝑊:ran 𝑊⟶dom 𝑊)
32 cycpmco2.j . . . . . . . . 9 (𝜑𝐽 ∈ ran 𝑊)
3331, 32ffvelrnd 6851 . . . . . . . 8 (𝜑 → (𝑊𝐽) ∈ dom 𝑊)
34 wrddm 13867 . . . . . . . . 9 (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
3511, 34syl 17 . . . . . . . 8 (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
3633, 35eleqtrd 2915 . . . . . . 7 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘𝑊)))
37 fzofzp1 13133 . . . . . . 7 ((𝑊𝐽) ∈ (0..^(♯‘𝑊)) → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
3836, 37syl 17 . . . . . 6 (𝜑 → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
3928, 38eqeltrid 2917 . . . . 5 (𝜑𝐸 ∈ (0...(♯‘𝑊)))
4011, 27, 39pfxf1 30618 . . . 4 (𝜑 → (𝑊 prefix 𝐸):dom (𝑊 prefix 𝐸)–1-1𝐷)
4115s1f1 30619 . . . 4 (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1𝐷)
42 s1rn 13952 . . . . . . 7 (𝐼𝐷 → ran ⟨“𝐼”⟩ = {𝐼})
4315, 42syl 17 . . . . . 6 (𝜑 → ran ⟨“𝐼”⟩ = {𝐼})
4443ineq2d 4188 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∩ {𝐼}))
45 pfxrn2 30616 . . . . . . . . 9 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
4611, 39, 45syl2anc 586 . . . . . . . 8 (𝜑 → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
4746ssrind 4211 . . . . . . 7 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
4814eldifbd 3948 . . . . . . . 8 (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
49 disjsn 4646 . . . . . . . 8 ((ran 𝑊 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ ran 𝑊)
5048, 49sylibr 236 . . . . . . 7 (𝜑 → (ran 𝑊 ∩ {𝐼}) = ∅)
5147, 50sseqtrd 4006 . . . . . 6 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅)
52 ss0 4351 . . . . . 6 ((ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅ → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
5351, 52syl 17 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
5444, 53eqtrd 2856 . . . 4 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = ∅)
551, 13, 16, 40, 41, 54ccatf1 30625 . . 3 (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩):dom ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)–1-1𝐷)
56 lencl 13882 . . . . 5 (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
57 nn0fz0 13004 . . . . . 6 ((♯‘𝑊) ∈ ℕ0 ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
5857biimpi 218 . . . . 5 ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
5911, 56, 583syl 18 . . . 4 (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
6011, 39, 59, 27swrdf1 30630 . . 3 (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩):dom (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)–1-1𝐷)
61 ccatrn 13942 . . . . . . 7 (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩))
6213, 16, 61syl2anc 586 . . . . . 6 (𝜑 → ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩))
6362ineq1d 4187 . . . . 5 (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
64 indir 4251 . . . . 5 ((ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
6563, 64syl6eq 2872 . . . 4 (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))))
66 fz0ssnn0 13001 . . . . . . . . . 10 (0...(♯‘𝑊)) ⊆ ℕ0
6766, 39sseldi 3964 . . . . . . . . 9 (𝜑𝐸 ∈ ℕ0)
68 pfxval 14034 . . . . . . . . 9 ((𝑊 ∈ Word 𝐷𝐸 ∈ ℕ0) → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
6911, 67, 68syl2anc 586 . . . . . . . 8 (𝜑 → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
7069rneqd 5807 . . . . . . 7 (𝜑 → ran (𝑊 prefix 𝐸) = ran (𝑊 substr ⟨0, 𝐸⟩))
7170ineq1d 4187 . . . . . 6 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72 0elfz 13003 . . . . . . . 8 (𝐸 ∈ ℕ0 → 0 ∈ (0...𝐸))
7367, 72syl 17 . . . . . . 7 (𝜑 → 0 ∈ (0...𝐸))
74 elfzuz3 12904 . . . . . . . 8 (𝐸 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ𝐸))
75 eluzfz1 12913 . . . . . . . 8 ((♯‘𝑊) ∈ (ℤ𝐸) → 𝐸 ∈ (𝐸...(♯‘𝑊)))
7639, 74, 753syl 18 . . . . . . 7 (𝜑𝐸 ∈ (𝐸...(♯‘𝑊)))
77 eluzfz2 12914 . . . . . . . 8 ((♯‘𝑊) ∈ (ℤ𝐸) → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
7839, 74, 773syl 18 . . . . . . 7 (𝜑 → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
7911, 73, 39, 27, 76, 78swrdrndisj 30631 . . . . . 6 (𝜑 → (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
8071, 79eqtrd 2856 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
81 incom 4177 . . . . . 6 (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩)
8243ineq2d 4188 . . . . . . 7 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}))
83 swrdrn2 30628 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
8411, 39, 59, 83syl3anc 1367 . . . . . . . . . 10 (𝜑 → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
8584ssrind 4211 . . . . . . . . 9 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
8685, 50sseqtrd 4006 . . . . . . . 8 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅)
87 ss0 4351 . . . . . . . 8 ((ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅ → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) = ∅)
8886, 87syl 17 . . . . . . 7 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) = ∅)
8982, 88eqtrd 2856 . . . . . 6 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = ∅)
9081, 89syl5eq 2868 . . . . 5 (𝜑 → (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
9180, 90uneq12d 4139 . . . 4 (𝜑 → ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = (∅ ∪ ∅))
92 unidm 4127 . . . . 5 (∅ ∪ ∅) = ∅
9392a1i 11 . . . 4 (𝜑 → (∅ ∪ ∅) = ∅)
9465, 91, 933eqtrd 2860 . . 3 (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
951, 18, 20, 55, 60, 94ccatf1 30625 . 2 (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)):dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))–1-1𝐷)
96 cycpmco2.1 . . . 4 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
97 ovexd 7190 . . . . . 6 (𝜑 → ((𝑊𝐽) + 1) ∈ V)
9828, 97eqeltrid 2917 . . . . 5 (𝜑𝐸 ∈ V)
99 splval 14112 . . . . 5 ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ ⟨“𝐼”⟩ ∈ Word 𝐷)) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
1003, 98, 98, 16, 99syl13anc 1368 . . . 4 (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
10196, 100syl5eq 2868 . . 3 (𝜑𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
102101dmeqd 5773 . . 3 (𝜑 → dom 𝑈 = dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
103 eqidd 2822 . . 3 (𝜑𝐷 = 𝐷)
104101, 102, 103f1eq123d 6607 . 2 (𝜑 → (𝑈:dom 𝑈1-1𝐷 ↔ (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)):dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))–1-1𝐷))
10595, 104mpbird 259 1 (𝜑𝑈:dom 𝑈1-1𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {crab 3142  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  {csn 4566  ⟨cop 4572  ⟨cotp 4574  ◡ccnv 5553  dom cdm 5554  ran crn 5555  ⟶wf 6350  –1-1→wf1 6351  –1-1-onto→wf1o 6353  ‘cfv 6354  (class class class)co 7155  0cc0 10536  1c1 10537   + caddc 10539  ℕ0cn0 11896  ℤ≥cuz 12242  ...cfz 12891  ..^cfzo 13032  ♯chash 13689  Word cword 13860   ++ cconcat 13921  ⟨“cs1 13948   substr csubstr 14001   prefix cpfx 14031   splice csplice 14110  Basecbs 16482  SymGrpcsymg 18494  toCycctocyc 30748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-ot 4575  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-fz 12892  df-fzo 13033  df-fl 13161  df-mod 13237  df-hash 13690  df-word 13861  df-concat 13922  df-s1 13949  df-substr 14002  df-pfx 14032  df-splice 14111  df-csh 14150  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-tset 16583  df-efmnd 18033  df-symg 18495  df-tocyc 30749 This theorem is referenced by:  cycpmco2lem4  30771  cycpmco2lem5  30772  cycpmco2lem6  30773  cycpmco2lem7  30774  cycpmco2  30775
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