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

Theorem cycpmco2f1 33144
Description: The word U used in cycpmco2 33153 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 4080 . . . . . 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 2737 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
74, 5, 6tocycf 33137 . . . . . . . . 9 (𝐷𝑉𝑀:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶(Base‘𝑆))
81, 7syl 17 . . . . . . . 8 (𝜑𝑀:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶(Base‘𝑆))
98fdmd 6746 . . . . . . 7 (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
103, 9eleqtrd 2843 . . . . . 6 (𝜑𝑊 ∈ {𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷})
112, 10sselid 3981 . . . . 5 (𝜑𝑊 ∈ Word 𝐷)
12 pfxcl 14715 . . . . 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 3963 . . . . 5 (𝜑𝐼𝐷)
1615s1cld 14641 . . . 4 (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
17 ccatcl 14612 . . . 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 14683 . . . 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 5914 . . . . . . . . 9 (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
23 eqidd 2738 . . . . . . . . 9 (𝑤 = 𝑊𝐷 = 𝐷)
2421, 22, 23f1eq123d 6840 . . . . . . . 8 (𝑤 = 𝑊 → (𝑤:dom 𝑤1-1𝐷𝑊:dom 𝑊1-1𝐷))
2524elrab 3692 . . . . . . 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 6872 . . . . . . . . . 10 (𝑊:dom 𝑊1-1𝐷𝑊:ran 𝑊1-1-onto→dom 𝑊)
30 f1of 6848 . . . . . . . . . 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 14559 . . . . . . . . 9 (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
3511, 34syl 17 . . . . . . . 8 (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
3633, 35eleqtrd 2843 . . . . . . 7 (𝜑 → (𝑊𝐽) ∈ (0..^(♯‘𝑊)))
37 fzofzp1 13803 . . . . . . 7 ((𝑊𝐽) ∈ (0..^(♯‘𝑊)) → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
3836, 37syl 17 . . . . . 6 (𝜑 → ((𝑊𝐽) + 1) ∈ (0...(♯‘𝑊)))
3928, 38eqeltrid 2845 . . . . 5 (𝜑𝐸 ∈ (0...(♯‘𝑊)))
4011, 27, 39pfxf1 32926 . . . 4 (𝜑 → (𝑊 prefix 𝐸):dom (𝑊 prefix 𝐸)–1-1𝐷)
4115s1f1 32927 . . . 4 (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1𝐷)
42 s1rn 14637 . . . . . . 7 (𝐼𝐷 → ran ⟨“𝐼”⟩ = {𝐼})
4315, 42syl 17 . . . . . 6 (𝜑 → ran ⟨“𝐼”⟩ = {𝐼})
4443ineq2d 4220 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 prefix 𝐸) ∩ {𝐼}))
45 pfxrn2 32924 . . . . . . . . 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 4244 . . . . . . 7 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
4814eldifbd 3964 . . . . . . . 8 (𝜑 → ¬ 𝐼 ∈ ran 𝑊)
49 disjsn 4711 . . . . . . . 8 ((ran 𝑊 ∩ {𝐼}) = ∅ ↔ ¬ 𝐼 ∈ ran 𝑊)
5048, 49sylibr 234 . . . . . . 7 (𝜑 → (ran 𝑊 ∩ {𝐼}) = ∅)
5147, 50sseqtrd 4020 . . . . . 6 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅)
52 ss0 4402 . . . . . 6 ((ran (𝑊 prefix 𝐸) ∩ {𝐼}) ⊆ ∅ → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
5351, 52syl 17 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ {𝐼}) = ∅)
5444, 53eqtrd 2777 . . . 4 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran ⟨“𝐼”⟩) = ∅)
551, 13, 16, 40, 41, 54ccatf1 32933 . . 3 (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩):dom ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)–1-1𝐷)
56 lencl 14571 . . . . 5 (𝑊 ∈ Word 𝐷 → (♯‘𝑊) ∈ ℕ0)
57 nn0fz0 13665 . . . . . 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 32941 . . 3 (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩):dom (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)–1-1𝐷)
61 ccatrn 14627 . . . . . . 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 4219 . . . . 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 4286 . . . . 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 2793 . . . 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 13662 . . . . . . . . . 10 (0...(♯‘𝑊)) ⊆ ℕ0
6766, 39sselid 3981 . . . . . . . . 9 (𝜑𝐸 ∈ ℕ0)
68 pfxval 14711 . . . . . . . . 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 5949 . . . . . . 7 (𝜑 → ran (𝑊 prefix 𝐸) = ran (𝑊 substr ⟨0, 𝐸⟩))
7170ineq1d 4219 . . . . . 6 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
72 0elfz 13664 . . . . . . . 8 (𝐸 ∈ ℕ0 → 0 ∈ (0...𝐸))
7367, 72syl 17 . . . . . . 7 (𝜑 → 0 ∈ (0...𝐸))
74 elfzuz3 13561 . . . . . . . 8 (𝐸 ∈ (0...(♯‘𝑊)) → (♯‘𝑊) ∈ (ℤ𝐸))
75 eluzfz1 13571 . . . . . . . 8 ((♯‘𝑊) ∈ (ℤ𝐸) → 𝐸 ∈ (𝐸...(♯‘𝑊)))
7639, 74, 753syl 18 . . . . . . 7 (𝜑𝐸 ∈ (𝐸...(♯‘𝑊)))
77 eluzfz2 13572 . . . . . . . 8 ((♯‘𝑊) ∈ (ℤ𝐸) → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
7839, 74, 773syl 18 . . . . . . 7 (𝜑 → (♯‘𝑊) ∈ (𝐸...(♯‘𝑊)))
7911, 73, 39, 27, 76, 78swrdrndisj 32942 . . . . . 6 (𝜑 → (ran (𝑊 substr ⟨0, 𝐸⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
8071, 79eqtrd 2777 . . . . 5 (𝜑 → (ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
81 incom 4209 . . . . . 6 (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩)
8243ineq2d 4220 . . . . . . 7 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}))
83 swrdrn2 32939 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝐷𝐸 ∈ (0...(♯‘𝑊)) ∧ (♯‘𝑊) ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
8411, 39, 59, 83syl3anc 1373 . . . . . . . . . 10 (𝜑 → ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ⊆ ran 𝑊)
8584ssrind 4244 . . . . . . . . 9 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ (ran 𝑊 ∩ {𝐼}))
8685, 50sseqtrd 4020 . . . . . . . 8 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ {𝐼}) ⊆ ∅)
87 ss0 4402 . . . . . . . 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 2777 . . . . . 6 (𝜑 → (ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∩ ran ⟨“𝐼”⟩) = ∅)
9081, 89eqtrid 2789 . . . . 5 (𝜑 → (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
9180, 90uneq12d 4169 . . . 4 (𝜑 → ((ran (𝑊 prefix 𝐸) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) ∪ (ran ⟨“𝐼”⟩ ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))) = (∅ ∪ ∅))
92 unidm 4157 . . . . 5 (∅ ∪ ∅) = ∅
9392a1i 11 . . . 4 (𝜑 → (∅ ∪ ∅) = ∅)
9465, 91, 933eqtrd 2781 . . 3 (𝜑 → (ran ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∩ ran (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)) = ∅)
951, 18, 20, 55, 60, 94ccatf1 32933 . 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 2845 . . . . 5 (𝜑𝐸 ∈ V)
99 splval 14789 . . . . 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 1374 . . . 4 (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
10196, 100eqtrid 2789 . . 3 (𝜑𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
102101dmeqd 5916 . . 3 (𝜑 → dom 𝑈 = dom (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
103 eqidd 2738 . . 3 (𝜑𝐷 = 𝐷)
104101, 102, 103f1eq123d 6840 . 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 1540  wcel 2108  {crab 3436  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cop 4632  cotp 4634  ccnv 5684  dom cdm 5685  ran crn 5686  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158  0cn0 12526  cuz 12878  ...cfz 13547  ..^cfzo 13694  chash 14369  Word cword 14552   ++ cconcat 14608  ⟨“cs1 14633   substr csubstr 14678   prefix cpfx 14708   splice csplice 14787  Basecbs 17247  SymGrpcsymg 19386  toCycctocyc 33126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-splice 14788  df-csh 14827  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-tset 17316  df-efmnd 18882  df-symg 19387  df-tocyc 33127
This theorem is referenced by:  cycpmco2lem4  33149  cycpmco2lem5  33150  cycpmco2lem6  33151  cycpmco2lem7  33152  cycpmco2  33153
  Copyright terms: Public domain W3C validator