WildClassification.lean`

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Dickson's classification: the wild case

This file classifies the finite subgroups G of PGLOf (K p) = PGL₂(𝔽̄_p) of order divisible by p (the wild case): the main theorem Dickson.classification_wild_slop shows any such G is

a semidirect product of an elementary abelian p-group by a cyclic group of order prime to p (the case where G fixes a point of ℙ¹(𝔽̄_p)), or
isomorphic to A₄, S₄ or A₅, or
isomorphic to PSL₂(𝔽_q) or PGL₂(𝔽_q) for some power q = p^m.

The proof combines the partition equation from FLT.Slop.PGL2.FiniteSubgroups.PartitionProof with the recognition theorems for A₅ (FLT.Slop.PGL2.FiniteSubgroups.RecognitionA5) and for PSL₂(𝔽_q)/PGL₂(𝔽_q) (FLT.Slop.PGL2.FiniteSubgroups.PSLRecognition and the field reconstruction in FLT.Slop.PGL2.FiniteSubgroups.FieldReconstruction).

Imports (7)

Imported by 2 files

def Dickson.dilationHomOfFix The group homomorphism from a finite subgroup H of PGLOf (K p) fixing infinity to (K p)ˣ, sending each element to its dilation parameter line 62 source
Source documentation
The group homomorphism from a finite subgroup H of PGLOf (K p) fixing infinity to (K p)ˣ, sending each element to its dilation parameter.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, OfNat.ofNat, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Eq, Dickson.ProjectiveLine, HSMul.hSMul, Dickson.infinity, MonoidHom and 1 more
Proof uses: MonoidHom.mk, OneHom.mk, Units.mk0, Dickson.dilationParam
56 instances, coercions and auxiliary constants left out.
```
noncomputable def dilationHomOfFix (H : Subgroup (PGLOf (K p))) [Finite H]
    (hH_fix : ∀ g : H, (g : PGLOf (K p)) • infinity p = infinity p) :
    H →* (K p)ˣ where
…
```
Mentions: Dickson.PGLOf, Dickson.K, Dickson.infinity
theorem Dickson.dilationHom_ker_order line 70 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, OfNat.ofNat, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Eq, Dickson.ProjectiveLine, HSMul.hSMul, Dickson.infinity, MonoidHom.ker and 4 more
Proof uses: Dickson.translationPGL, Exists, Iff.mp, Dickson.dilationParam, Dickson.dilationParam_eq_one_iff, congr_arg, MonoidHom, HPow.hPow, Nat.recAux, Eq.mpr, id, congrArg, pow_zero, zero_smul, Eq.symm, Dickson.translationPGL_zero, Eq.refl, HAdd.hAdd, HMul.hMul, pow_succ and 30 more
107 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
theorem Dickson.fixes_infinity_coprime_isCyclic line 90 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, OfNat.ofNat, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Eq, Dickson.ProjectiveLine, HSMul.hSMul, Dickson.infinity, Nat.Coprime and 2 more
Proof uses: MonoidHom, Units, Dickson.dilationHomOfFix, MonoidHom.ker, Bot.bot, Iff.mpr, LE.le, eq_bot_iff, Subgroup.mem_bot, orderOf, Dvd.dvd, id, orderOf_dvd_natCard, Dickson.dilationHom_ker_order, Nat.gcd, Nat.Coprime.gcd_eq_one, Nat.Coprime.coprime_dvd_left, Eq.mpr, congrArg, Eq.symm and 33 more
84 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
abbrev Dickson.partitionTerm A single summand 1 / sᵢ * (1 - 1 / zᵢ) appearing in the class-equation sum line 135 source
Source documentation
A single summand 1 / sᵢ * (1 - 1 / zᵢ) appearing in the class-equation sum.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
Proof uses: HMul.hMul, HDiv.hDiv, OfNat.ofNat, HSub.hSub
9 instances, coercions and auxiliary constants left out.
```
abbrev partitionTerm (si zi : ℕ) : ℚ := 1 / (si : ℚ) * (1 - 1 / (zi : ℚ))
```
theorem Dickson.sylow_comm_exp_p line 139 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Sylow, And, Eq, HMul.hMul and 1 more
Proof uses: Subgroup.map, Subgroup.subtype, Finite.of_surjective, Subtype.mk, Subgroup.mem_map_of_mem, Subtype.property, Exists, MonoidHom, Iff.mp, Subgroup.mem_map, Exists.intro, rfl, IsPGroup, IsPGroup.map, Sylow.isPGroup', dite, Nontrivial, Dvd.dvd, Monoid.exponent, ExistsUnique and 21 more
72 instances, coercions and auxiliary constants left out.

Tactic trace: 41 steps
Open to load the checked before-and-after goal states.
theorem Dickson.group_fixes_sylow_point line 179 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Dvd.dvd, Nat.card, Sylow, Subgroup.Normal and 4 more
Proof uses: Subgroup.map, Subgroup.subtype, And, Exists.choose, Dickson.sylow_unique_fixedPoint, Exists.choose_spec, MonoidHom, Exists, Iff.mp, Subgroup.mem_map, HMul.hMul, Inv.inv, Subgroup.mem_map_of_mem, Eq.rec, Subgroup.Normal.conj_mem, inv_inv, Eq.mpr, id, congrArg, mul_assoc and 6 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
theorem Dickson.fixes_point_coprime_isCyclic line 198 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, OfNat.ofNat, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Dickson.ProjectiveLine, Eq, HSMul.hSMul, Nat.Coprime, Nat.card and 1 more
Proof uses: Dickson.infinity, Exists, Dickson.exists_smul_eq_infinity, MulEquiv, Subgroup.map, MonoidHom, MulAut, MulAut.conj, MulEquiv.subgroupMap, MulEquiv.toMonoidHom, Finite.of_equiv, MulEquiv.toEquiv, And, Subtype.mk, Iff.mp, Subgroup.mem_map, HMul.hMul, Inv.inv, Eq.mpr, id and 10 more
70 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
theorem Dickson.complement_order_coprime line 218 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, OfNat.ofNat, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Sylow, Subgroup.IsComplement', Nat.Coprime, Nat.card
Proof uses: Eq, HMul.hMul, HPow.hPow, Finsupp, Nat.factorization, Eq.symm, Subgroup.IsComplement'.card_mul_card, Eq.mp, congrArg, Sylow.card_eq_multiplicity, Nat.Coprime.symm, Iff.mpr, Not, Dvd.dvd, Nat.Prime.coprime_iff_not_dvd, Fact.out, Nat.pow_succ_factorization_not_dvd, LT.lt.ne', Nat.card_pos, Eq.rec and 1 more
51 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Dickson.complement_isCyclic line 230 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Dvd.dvd, Nat.card, Sylow, Subgroup.Normal and 2 more
Proof uses: Subgroup.map, Subgroup.subtype, MulEquiv, Subgroup.equivMapOfInjective, Subtype.coe_injective, Finite.of_equiv, MulEquiv.toEquiv, Dickson.ProjectiveLine, Dickson.sylowFixedPoint, Eq, HSMul.hSMul, And, MonoidHom, Exists, Subtype.mk, Iff.mp, Subgroup.mem_map, Dickson.group_fixes_sylow_point, Iff.mpr, MulEquiv.isCyclic and 5 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
theorem Dickson.mul_elem_abelian_iso line 247 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, CommGroup, Finite, Eq, HPow.hPow, OfNat.ofNat, Exists, Nonempty, MulEquiv, Multiplicative, Fin, ZMod
Proof uses: Module, Additive, AddCommGroup.zmodModule, Equiv, Additive.toMul, Exists.intro, Module.finrank, Nonempty.intro, AddEquiv, AddEquiv.toMultiplicative, LinearEquiv.toAddEquiv, RingHom.id, Module.Basis.equivFun, Module.finBasis
79 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Dickson.semidirect_of_complement_iso line 254 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Subgroup, Subgroup.Normal, Subgroup.IsComplement', MulEquiv, Subtype, Membership.mem, Exists, MonoidHom, MulAut, Nonempty, SemidirectProduct
Proof uses: Exists.intro, MonoidHom.comp, MulAut.congr, Subgroup.normalizer, Subgroup.normalizerMonoidHom, Subgroup.inclusion, Eq.rec, Top.top, Eq, LE.le, le_top, Eq.symm, Subgroup.normalizer_eq_top, MulEquiv.symm, Nonempty.intro, MulEquiv.trans, SemidirectProduct.mulEquivSubgroup, SemidirectProduct.congr'
37 instances, coercions and auxiliary constants left out.
theorem Dickson.branch1_semidirect line 263 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Sylow, Subgroup.Normal, Dvd.dvd, Nat.card and 11 more
Proof uses: Subgroup.IsComplement', Subgroup.exists_right_complement'_of_coprime, Sylow.card_coprime_index, And, Eq, HMul.hMul, Dickson.sylow_comm_exp_p, CommGroup, Group, inferInstance, CommGroup.mk, Dickson.mul_elem_abelian_iso, IsCyclic, Dickson.complement_isCyclic, Dickson.complement_order_coprime, Nonempty.intro, MulEquiv.symm, zmodCyclicMulEquiv, Dickson.semidirect_of_complement_iso, Eq.mpr and 37 more
131 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
theorem Dickson.branch2a_constraints line 296 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, GE.ge, OfNat.ofNat, GT.gt, Or, Eq, Rat, HSub.hSub, HDiv.hDiv, Finset.sum, Finset.univ, Dickson.partitionTerm, And
Proof uses: LE.le, Eq.mpr, Inv.inv, id, congrArg, one_div, inv_anti₀, zero_lt_three, Iff.mpr, Nat.cast_le, LT.lt, one_div_pos, Nat.cast_pos, zero_lt_two, HMul.hMul, Dickson.partitionTerm.eq_1, le_of_not_gt, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HAdd.hAdd and 123 more
165 instances, coercions and auxiliary constants left out.

Tactic trace: 67 steps
Open to load the checked before-and-after goal states.
theorem Dickson.recognition_A4 line 346 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, OfNat.ofNat, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Eq, Nat.card, Fintype.card, Sylow, Dickson.instFintypeSylow and 4 more
Proof uses: LE.le, SetLike.ext', Set.eq_of_subset_of_ncard_le, Eq.ge, Set.toFinite, Set, Eq.trans, HPow.hPow, Finsupp, Nat.factorization, Sylow.card_eq_multiplicity, Eq.mpr, id, congrArg, HMul.hMul, Dickson.Nat.factorization_eq_one, rfl, Nat.prime_three, Mathlib.Meta.NormNum.isNat_dvd_false, Mathlib.Meta.NormNum.isNat_ofNat and 68 more
105 instances, coercions and auxiliary constants left out.

Tactic trace: 48 steps
Open to load the checked before-and-after goal states.
theorem Dickson.pgl_unique_index_two_subgroup line 390 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, GE.ge, HPow.hPow, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, LE.le, MonoidHom.range, GaloisField, Dickson.pglMap, Dickson.galoisFieldRingHom and 11 more
Proof uses: MonoidHom, Subgroup.comap, MulEquiv.trans, Subgroup.map, Subgroup.equivMapOfInjective, Dickson.pglMap_injective, RingHom.injective, MulEquiv.subgroupCongr, Subgroup.ext, Eq.mpr, Iff, Exists, And, id, congrArg, propext, Subgroup.mem_map, Iff.intro, Exists.intro, And.intro and 63 more
135 instances, coercions and auxiliary constants left out.

Tactic trace: 36 steps
Open to load the checked before-and-after goal states.
theorem Dickson.m_ge_one_of_pow_ge_three line 428 source
standard axioms: propext what this means
Statement uses: Nat, GE.ge, HPow.hPow, OfNat.ofNat
Proof uses: Nat.casesAuxOn, Eq, absurd, Nat.not_le_of_gt, Nat.le.step, Nat.succ, Nat.le.refl, Eq.symm, HAdd.hAdd, Nat.succ_pos, Eq.refl
15 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Dickson.recognition_PSL_iso line 433 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, GE.ge, HPow.hPow, Eq, Nat.card and 8 more
Proof uses: Dickson.m_ge_one_of_pow_ge_three, Nat.le_trans, Nat.succ, Nat.le.step, Nat.le.refl, LE.le, Subgroup.map, MulEquiv.toMonoidHom, MonoidHom, MulAut, MulAut.conj, MonoidHom.range, Dickson.pglMap, Dickson.galoisFieldRingHom, Exists, Dickson.psl_G_le_pgl_range_from_orbit, Dickson.pgl_unique_index_two_subgroup, Eq.mpr, id, congrArg and 8 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem Dickson.recognition_PGL_iso line 443 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, GE.ge, HPow.hPow, Eq, Nat.card and 6 more
Proof uses: Dickson.m_ge_one_of_pow_ge_three, MonoidHom.range, Dickson.pglMap, Dickson.galoisFieldRingHom, Set.Finite.to_subtype, Set.range, MonoidHom, Set.finite_range, Subgroup.map, MulEquiv.toMonoidHom, MulAut, MulAut.conj, Exists, Dickson.pgl_subgroups_conjugate, Eq.trans, Nat.card_congr, Equiv.symm, Equiv.ofInjective, Dickson.pglMap_injective, RingHom.injective and 18 more
86 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
theorem Dickson.recognition_PSL_pm3 line 459 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Eq, Nat.card, Fintype.card, Sylow and 6 more
Proof uses: Equiv.Perm, alternatingGroup, Dickson.recognition_A4, Dickson.instFactPrimeOfNatNat_fLT_1, A4_iso_PSL2_GF31, Nonempty.intro, MulEquiv.trans, Eq.symm
55 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
theorem Dickson.sylow_count_of_order_12 line 470 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, OfNat.ofNat, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Eq, Nat.card, GT.gt, Fintype.card, Sylow and 1 more
Proof uses: Dvd.dvd, HasQuotient.Quotient, HDiv.hDiv, HPow.hPow, Finsupp, Nat.factorization, Eq.mpr, Eq.refl, Sylow.card_eq_multiplicity, id, congrArg, Eq.symm, HMul.hMul, Subgroup.card_eq_card_quotient_mul_card_subgroup, Nat.mul_div_cancel, Nat.card_pos, Subgroup.normalizer, Nat.card_eq_fintype_card, Nat.card_congr, Sylow.equivQuotientNormalizer and 39 more
71 instances, coercions and auxiliary constants left out.

Tactic trace: 33 steps
Open to load the checked before-and-after goal states.
theorem Dickson.recognition_PSL_general line 504 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, GE.ge, HPow.hPow, Eq, Nat.card and 12 more
Proof uses: dite, Dickson.recognition_PSL_iso, Not, Fact.out, LE.le, LE.le.trans, Nat.le_self_pow, Ne, Not.intro, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, LT.lt, HAdd.hAdd, Neg.neg, congrArg, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance and 100 more
152 instances, coercions and auxiliary constants left out.

Tactic trace: 29 steps
Open to load the checked before-and-after goal states.
theorem Dickson.partition_equation line 530 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Dvd.dvd, Nat.card, Sylow, Fintype.card and 15 more
Proof uses: Dickson.normalizerQuotient, Dickson.nonsplit_torus_divides_geo, Exists.intro, And.intro, rfl, Dickson.sylow_card_ge_3, Nat.div_pos, Subgroup.normalizer, Subgroup.card_le_of_le, Subgroup.le_normalizer, Nat.card_pos, Dickson.normalizer_complement_divides_main, Or.inr, Eq.refl, Dickson.nonsplit_torus_ge_two, Dickson.group_order_gt_normalizer, LE.le, Ne, Iff.mpr, Nat.cast_ne_zero and 111 more
160 instances, coercions and auxiliary constants left out.

Tactic trace: 61 steps
Open to load the checked before-and-after goal states.
theorem Dickson.dickson_branch2_z1_eq_1 line 593 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Dvd.dvd, Nat.card, Sylow, Fintype.card and 29 more
Proof uses: Eq.mpr, id, congrArg, ite_eq_left, rfl, Dickson.partitionTerm.eq_1, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Meta.NormNum.isNat_mul, Eq.refl, Mathlib.Meta.NormNum.IsNNRat.to_isNat, Mathlib.Meta.NormNum.isNNRat_div, Mathlib.Meta.NormNum.IsNat.to_isNNRat, Inv.inv, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.isNNRat_inv_pos, Mathlib.Meta.NormNum.isNat_natCast, Mathlib.Meta.NormNum.IsInt.to_isNat, Mathlib.Meta.NormNum.isInt_sub, Int.ofNat and 103 more
239 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.
theorem Dickson.dickson_branch2_z1_ge_2 line 639 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Dvd.dvd, Nat.card, Sylow, Fintype.card and 21 more
Proof uses: Dickson.normalizerQuotient, HDiv.hDiv, HAdd.hAdd, Dickson.branch2_params, Finsupp, Nat.factorization, Sylow.card_eq_multiplicity, Iff.mp, Nat.prime_dvd_prime_iff_eq, Fact.out, Nat.prime_three, Eq.rec, dvd_pow_self, Eq.mp, congrArg, pow_zero, Eq.symm, And.left, Or.inr, Or.inl and 125 more
203 instances, coercions and auxiliary constants left out.

Tactic trace: 72 steps
Open to load the checked before-and-after goal states.
theorem Dickson.classification_wild_slop line 694 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fact, Nat.Prime, GT.gt, OfNat.ofNat, Subgroup, Dickson.PGLOf, Dickson.K, ZMod, Matrix.GeneralLinearGroup, Fin, Matrix, Subgroup.center, Finite, Subtype, Membership.mem, Dvd.dvd, Nat.card, Or, Exists and 18 more
Proof uses: Sylow, inferInstance, dite, Fintype.card, Dickson.instFintypeSylow, Subsingleton, Iff.mp, LE.le, Fintype.card_le_one_iff_subsingleton, le_of_eq, Or.inl, Dickson.branch1_semidirect, Sylow.normal_of_subsingleton, Not, lt_of_le_of_ne, Iff.mpr, LT.lt, Fintype.card_pos_iff, Ne.symm, HMul.hMul and 10 more
127 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.