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Theorem cycpmco2f1 33127
Description: The word U used in cycpmco2 33136 is injective, so it can represent a cycle and form a cyclic permutation (𝑀𝑈). (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 4090 . . . . . 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 2735 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
74, 5, 6tocycf 33120 . . . . . . . . 9 (𝐷𝑉𝑀:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶(Base‘𝑆))
81, 7syl 17 . . . . . . . 8 (𝜑𝑀:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶(Base‘𝑆))
98fdmd 6747 . . . . . . 7 (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
103, 9eleqtrd 2841 . . . . . 6 (𝜑𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
112, 10sselid 3993 . . . . 5 (𝜑𝑊 ∈ Word 𝐷)
12 pfxcl 14712 . . . . 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 3975 . . . . 5 (𝜑𝐼𝐷)
1615s1cld 14638 . . . 4 (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
17 ccatcl 14609 . . . 4 (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
1813, 16, 17syl2anc 584 . . 3 (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
19 swrdcl 14680 . . . 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 5917 . . . . . . . . 9 (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
23 eqidd 2736 . . . . . . . . 9 (𝑤 = 𝑊𝐷 = 𝐷)
2421, 22, 23f1eq123d 6841 . . . . . . . 8 (𝑤 = 𝑊 → (𝑤:dom 𝑤1-1𝐷𝑊:dom 𝑊1-1𝐷))
2524elrab 3695 . . . . . . 7 (𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷} ↔ (𝑊 ∈ Word 𝐷𝑊:dom 𝑊1-1𝐷))
2610, 25sylib 218 . . . . . 6 (𝜑 → (𝑊 ∈ Word 𝐷𝑊:dom 𝑊1-1𝐷))
2726simprd 495 . . . . 5 (𝜑𝑊:dom 𝑊1-1𝐷)
28 cycpmco2.e . . . . . 6 𝐸 = ((𝑊𝐽) + 1)
29 f1cnv 6873 . . . . . . . . . 10 (𝑊:dom 𝑊1-1𝐷𝑊:ran 𝑊1-1-onto→dom 𝑊)
30 f1of 6849 . . . . . . . . . 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, 32ffvelcdmd 7105 . . . . . . . 8 (𝜑 → (𝑊𝐽) ∈ dom 𝑊)
34 wrddm 14556 . . . . . . . . 9 (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
3511, 34syl 17 . . . . . . . 8 (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
3633, 35eleqtrd 2841 . . . . . . 7 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘𝑊)))
37 fzofzp1 13800 . . . . . . 7 ((𝑊𝐽) ∈ (0..^(♯‘𝑊)) → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
3836, 37syl 17 . . . . . 6 (𝜑 → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
3928, 38eqeltrid 2843 . . . . 5 (𝜑𝐸 ∈ (0...(♯‘𝑊)))
4011, 27, 39pfxf1 32911 . . . 4 (𝜑 → (𝑊 prefix 𝐸):dom (𝑊 prefix 𝐸)–1-1𝐷)
4115s1f1 32912 . . . 4 (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1𝐷)
42 s1rn 14634 . . . . . . 7 (𝐼𝐷 → ran ⟨“𝐼”⟩ = {𝐼})
4315, 42syl 17 . . . . . 6 (𝜑 → ran ⟨“𝐼”⟩ = {𝐼})
4443ineq2d 4228 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∩ {𝐼}))
45 pfxrn2 32909 . . . . . . . . 9 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
4611, 39, 45syl2anc 584 . . . . . . . 8 (𝜑 → ran (𝑊 prefix 𝐸) ⊆ ran 𝑊)
4746ssrind 4252 . . . . . . 7 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
4814eldifbd 3976 . . . . . . . 8 (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
49 disjsn 4716 . . . . . . . 8 ((ran 𝑊 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ ran 𝑊)
5048, 49sylibr 234 . . . . . . 7 (𝜑 → (ran 𝑊 ∩ {𝐼}) = ∅)
5147, 50sseqtrd 4036 . . . . . 6 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅)
52 ss0 4408 . . . . . 6 ((ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅ → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
5351, 52syl 17 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
5444, 53eqtrd 2775 . . . 4 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = ∅)
551, 13, 16, 40, 41, 54ccatf1 32918 . . 3 (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩):dom ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)–1-1𝐷)
56 lencl 14568 . . . . 5 (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
57 nn0fz0 13662 . . . . . 6 ((♯‘𝑊) ∈ ℕ0 ↔ (♯‘𝑊) ∈ (0...(♯‘𝑊)))
5857biimpi 216 . . . . 5 ((♯‘𝑊) ∈ ℕ0 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
5911, 56, 583syl 18 . . . 4 (𝜑 → (♯‘𝑊) ∈ (0...(♯‘𝑊)))
6011, 39, 59, 27swrdf1 32926 . . 3 (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩):dom (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)–1-1𝐷)
61 ccatrn 14624 . . . . . . 7 (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩))
6213, 16, 61syl2anc 584 . . . . . 6 (𝜑 → ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∪ ran ⟨“𝐼”⟩))
6362ineq1d 4227 . . . . 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 4292 . . . . 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, 64eqtrdi 2791 . . . 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 13659 . . . . . . . . . 10 (0...(♯‘𝑊)) ⊆ ℕ0
6766, 39sselid 3993 . . . . . . . . 9 (𝜑𝐸 ∈ ℕ0)
68 pfxval 14708 . . . . . . . . 9 ((𝑊 ∈ Word 𝐷𝐸 ∈ ℕ0) → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
6911, 67, 68syl2anc 584 . . . . . . . 8 (𝜑 → (𝑊 prefix 𝐸) = (𝑊 substr ⟨0, 𝐸⟩))
7069rneqd 5952 . . . . . . 7 (𝜑 → ran (𝑊 prefix 𝐸) = ran (𝑊 substr ⟨0, 𝐸⟩))
7170ineq1d 4227 . . . . . 6 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72 0elfz 13661 . . . . . . . 8 (𝐸 ∈ ℕ0 → 0 ∈ (0...𝐸))
7367, 72syl 17 . . . . . . 7 (𝜑 → 0 ∈ (0...𝐸))
74 elfzuz3 13558 . . . . . . . 8 (𝐸 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ𝐸))
75 eluzfz1 13568 . . . . . . . 8 ((♯‘𝑊) ∈ (ℤ𝐸) → 𝐸 ∈ (𝐸...(♯‘𝑊)))
7639, 74, 753syl 18 . . . . . . 7 (𝜑𝐸 ∈ (𝐸...(♯‘𝑊)))
77 eluzfz2 13569 . . . . . . . 8 ((♯‘𝑊) ∈ (ℤ𝐸) → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
7839, 74, 773syl 18 . . . . . . 7 (𝜑 → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
7911, 73, 39, 27, 76, 78swrdrndisj 32927 . . . . . 6 (𝜑 → (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
8071, 79eqtrd 2775 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
81 incom 4217 . . . . . 6 (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩)
8243ineq2d 4228 . . . . . . 7 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}))
83 swrdrn2 32924 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
8411, 39, 59, 83syl3anc 1370 . . . . . . . . . 10 (𝜑 → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
8584ssrind 4252 . . . . . . . . 9 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
8685, 50sseqtrd 4036 . . . . . . . 8 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅)
87 ss0 4408 . . . . . . . 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 2775 . . . . . 6 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = ∅)
9081, 89eqtrid 2787 . . . . 5 (𝜑 → (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
9180, 90uneq12d 4179 . . . 4 (𝜑 → ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = (∅ ∪ ∅))
92 unidm 4167 . . . . 5 (∅ ∪ ∅) = ∅
9392a1i 11 . . . 4 (𝜑 → (∅ ∪ ∅) = ∅)
9465, 91, 933eqtrd 2779 . . 3 (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
951, 18, 20, 55, 60, 94ccatf1 32918 . 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 7466 . . . . . 6 (𝜑 → ((𝑊𝐽) + 1) ∈ V)
9828, 97eqeltrid 2843 . . . . 5 (𝜑𝐸 ∈ V)
99 splval 14786 . . . . 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 1371 . . . 4 (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
10196, 100eqtrid 2787 . . 3 (𝜑𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
102101dmeqd 5919 . . 3 (𝜑 → dom 𝑈 = dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
103 eqidd 2736 . . 3 (𝜑𝐷 = 𝐷)
104101, 102, 103f1eq123d 6841 . 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 257 1 (𝜑𝑈:dom 𝑈1-1𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631  cop 4637  cotp 4639  ccnv 5688  dom cdm 5689  ran crn 5690  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   + caddc 11156  0cn0 12524  cuz 12876  ...cfz 13544  ..^cfzo 13691  chash 14366  Word cword 14549   ++ cconcat 14605  ⟨“cs1 14630   substr csubstr 14675   prefix cpfx 14705   splice csplice 14784  Basecbs 17245  SymGrpcsymg 19401  toCycctocyc 33109
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-splice 14785  df-csh 14824  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-tset 17317  df-efmnd 18895  df-symg 19402  df-tocyc 33110
This theorem is referenced by:  cycpmco2lem4  33132  cycpmco2lem5  33133  cycpmco2lem6  33134  cycpmco2lem7  33135  cycpmco2  33136
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