plotEncoding_image_eq_nonzero

formal source

 160
 161/-- The image of `plotEncoding` is exactly the seven non-zero
 162    vectors of `F2Power 3`. -/
 163theorem plotEncoding_image_eq_nonzero :
 164    Finset.univ.image plotEncoding =
 165      Finset.univ.filter (fun v : F2Power 3 => v ≠ 0) := by
 166  apply Finset.eq_of_subset_of_card_le
 167  · intro v hv
 168    rcases Finset.mem_image.mp hv with ⟨p, _, hp⟩
 169    rw [Finset.mem_filter, ← hp]
 170    exact ⟨Finset.mem_univ _, plotEncoding_image_nonzero p⟩
 171  · rw [F2Power.nonzero_card_three]
 172    have : (Finset.univ.image plotEncoding).card =
 173           Fintype.card BookerPlotFamily := by
 174      rw [Finset.card_image_of_injective _ plotEncoding_injective]
 175      rfl
 176    rw [this, BookerPlotFamily.card_eq_seven]
 177
 178/-! ## §3. The count theorem (replaces the asserted `:= 7`) -/
 179
 180/-- The number of non-trivial 1-dimensional subgroups of
 181    `Q₃ = F2Power 3` is `2 ^ 3 - 1 = 7`. This is the actual count
 182    theorem, chained off `F2Power.nonzero_card_three`. -/
 183theorem Q3_nontrivial_subgroup_count :
 184    (Finset.univ.filter (fun v : F2Power 3 => v ≠ 0)).card = 7 :=
 185  F2Power.nonzero_card_three
 186
 187/-- The number of Booker plot families equals the non-zero cardinality
 188    of `F2Power 3`. -/
 189theorem booker_count_eq_F2Power3_nonzero :
 190    Fintype.card BookerPlotFamily =
 191      (Finset.univ.filter (fun v : F2Power 3 => v ≠ 0)).card := by
 192  rw [BookerPlotFamily.card_eq_seven, Q3_nontrivial_subgroup_count]
 193