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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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