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Theorem trsp2cyc 30783
 Description: Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.)
Hypotheses
Ref Expression
trsp2cyc.t 𝑇 = ran (pmTrsp‘𝐷)
trsp2cyc.c 𝐶 = (toCyc‘𝐷)
Assertion
Ref Expression
trsp2cyc ((𝐷𝑉𝑃𝑇) → ∃𝑖𝐷𝑗𝐷 (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
Distinct variable groups:   𝐷,𝑖,𝑗   𝑃,𝑖,𝑗   𝑇,𝑖,𝑗   𝑖,𝑉,𝑗
Allowed substitution hints:   𝐶(𝑖,𝑗)

Proof of Theorem trsp2cyc
Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 768 . . . . . . 7 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o})
2 breq1 5050 . . . . . . . 8 (𝑦 = 𝑝 → (𝑦 ≈ 2o𝑝 ≈ 2o))
32elrab 3665 . . . . . . 7 (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↔ (𝑝 ∈ 𝒫 𝐷𝑝 ≈ 2o))
41, 3sylib 221 . . . . . 6 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → (𝑝 ∈ 𝒫 𝐷𝑝 ≈ 2o))
54simprd 499 . . . . 5 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝 ≈ 2o)
6 en2 8738 . . . . 5 (𝑝 ≈ 2o → ∃𝑖𝑗 𝑝 = {𝑖, 𝑗})
75, 6syl 17 . . . 4 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → ∃𝑖𝑗 𝑝 = {𝑖, 𝑗})
84simpld 498 . . . . . . . . . 10 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝 ∈ 𝒫 𝐷)
98elpwid 4531 . . . . . . . . 9 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → 𝑝𝐷)
109adantr 484 . . . . . . . 8 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝𝐷)
11 vex 3482 . . . . . . . . . 10 𝑖 ∈ V
1211prid1 4679 . . . . . . . . 9 𝑖 ∈ {𝑖, 𝑗}
13 simpr 488 . . . . . . . . 9 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 = {𝑖, 𝑗})
1412, 13eleqtrrid 2923 . . . . . . . 8 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑖𝑝)
1510, 14sseldd 3952 . . . . . . 7 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑖𝐷)
16 vex 3482 . . . . . . . . . 10 𝑗 ∈ V
1716prid2 4680 . . . . . . . . 9 𝑗 ∈ {𝑖, 𝑗}
1817, 13eleqtrrid 2923 . . . . . . . 8 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑗𝑝)
1910, 18sseldd 3952 . . . . . . 7 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑗𝐷)
205adantr 484 . . . . . . . . . 10 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 ≈ 2o)
2113, 20eqbrtrrd 5071 . . . . . . . . 9 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → {𝑖, 𝑗} ≈ 2o)
22 pr2ne 9416 . . . . . . . . . 10 ((𝑖𝐷𝑗𝐷) → ({𝑖, 𝑗} ≈ 2o𝑖𝑗))
2322biimpa 480 . . . . . . . . 9 (((𝑖𝐷𝑗𝐷) ∧ {𝑖, 𝑗} ≈ 2o) → 𝑖𝑗)
2415, 19, 21, 23syl21anc 836 . . . . . . . 8 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑖𝑗)
25 simplr 768 . . . . . . . . . 10 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
26 simp-4l 782 . . . . . . . . . . 11 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝐷𝑉)
27 eqid 2824 . . . . . . . . . . . 12 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
2827pmtrval 18568 . . . . . . . . . . 11 ((𝐷𝑉𝑝𝐷𝑝 ≈ 2o) → ((pmTrsp‘𝐷)‘𝑝) = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
2926, 10, 20, 28syl3anc 1368 . . . . . . . . . 10 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → ((pmTrsp‘𝐷)‘𝑝) = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
3013fveq2d 6655 . . . . . . . . . 10 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → ((pmTrsp‘𝐷)‘𝑝) = ((pmTrsp‘𝐷)‘{𝑖, 𝑗}))
3125, 29, 303eqtr2d 2865 . . . . . . . . 9 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑃 = ((pmTrsp‘𝐷)‘{𝑖, 𝑗}))
32 trsp2cyc.c . . . . . . . . . 10 𝐶 = (toCyc‘𝐷)
3332, 26, 15, 19, 24, 27cycpm2tr 30779 . . . . . . . . 9 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → (𝐶‘⟨“𝑖𝑗”⟩) = ((pmTrsp‘𝐷)‘{𝑖, 𝑗}))
3431, 33eqtr4d 2862 . . . . . . . 8 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))
3524, 34jca 515 . . . . . . 7 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
3615, 19, 35jca31 518 . . . . . 6 (((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ∧ 𝑝 = {𝑖, 𝑗}) → ((𝑖𝐷𝑗𝐷) ∧ (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))))
3736ex 416 . . . . 5 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → (𝑝 = {𝑖, 𝑗} → ((𝑖𝐷𝑗𝐷) ∧ (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))))
38372eximdv 1921 . . . 4 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → (∃𝑖𝑗 𝑝 = {𝑖, 𝑗} → ∃𝑖𝑗((𝑖𝐷𝑗𝐷) ∧ (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))))
397, 38mpd 15 . . 3 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → ∃𝑖𝑗((𝑖𝐷𝑗𝐷) ∧ (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))))
40 r2ex 3295 . . 3 (∃𝑖𝐷𝑗𝐷 (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)) ↔ ∃𝑖𝑗((𝑖𝐷𝑗𝐷) ∧ (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))))
4139, 40sylibr 237 . 2 ((((𝐷𝑉𝑃𝑇) ∧ 𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}) ∧ 𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) → ∃𝑖𝐷𝑗𝐷 (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
42 simpr 488 . . . 4 ((𝐷𝑉𝑃𝑇) → 𝑃𝑇)
43 trsp2cyc.t . . . . 5 𝑇 = ran (pmTrsp‘𝐷)
4427pmtrfval 18567 . . . . . . 7 (𝐷𝑉 → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
4544adantr 484 . . . . . 6 ((𝐷𝑉𝑃𝑇) → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
4645rneqd 5789 . . . . 5 ((𝐷𝑉𝑃𝑇) → ran (pmTrsp‘𝐷) = ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
4743, 46syl5eq 2871 . . . 4 ((𝐷𝑉𝑃𝑇) → 𝑇 = ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
4842, 47eleqtrd 2918 . . 3 ((𝐷𝑉𝑃𝑇) → 𝑃 ∈ ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
49 eqid 2824 . . . . 5 (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
5049elrnmpt 5809 . . . 4 (𝑃𝑇 → (𝑃 ∈ ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ↔ ∃𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
5150adantl 485 . . 3 ((𝐷𝑉𝑃𝑇) → (𝑃 ∈ ran (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))) ↔ ∃𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
5248, 51mpbid 235 . 2 ((𝐷𝑉𝑃𝑇) → ∃𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2o}𝑃 = (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
5341, 52r19.29a 3281 1 ((𝐷𝑉𝑃𝑇) → ∃𝑖𝐷𝑗𝐷 (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ≠ wne 3013  ∃wrex 3133  {crab 3136   ∖ cdif 3915   ⊆ wss 3918  ifcif 4448  𝒫 cpw 4520  {csn 4548  {cpr 4550  ∪ cuni 4819   class class class wbr 5047   ↦ cmpt 5127  ran crn 5537  ‘cfv 6336  2oc2o 8079   ≈ cen 8489  ⟨“cs2 14192  pmTrspcpmtr 18558  toCycctocyc 30766 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599  ax-pre-sup 10600 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-card 9352  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-div 11283  df-nn 11624  df-2 11686  df-n0 11884  df-xnn0 11954  df-z 11968  df-uz 12230  df-rp 12376  df-fz 12884  df-fzo 13027  df-fl 13155  df-mod 13231  df-hash 13685  df-word 13856  df-concat 13912  df-s1 13939  df-substr 13992  df-pfx 14022  df-csh 14140  df-s2 14199  df-pmtr 18559  df-tocyc 30767 This theorem is referenced by:  cyc3genpm  30812
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