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[Merged by Bors] - chore: golf FiniteDimensional.isROrC_to_real #9921

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[Merged by Bors] - chore: golf FiniteDimensional.isROrC_to_real #9921

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chore: golf `FiniteDimensional.isROrC_to_real`
j-loreaux Jan 22, 2024
77427f8
use `suffices`
j-loreaux Jan 23, 2024
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10 changes: 2 additions & 8 deletions Mathlib/Data/IsROrC/Lemmas.lean
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Expand Up @@ -38,14 +38,8 @@ This instance generates a type-class problem with a metavariable `?m` that shoul
/-- An `IsROrC` field is finite-dimensional over `ℝ`, since it is spanned by `{1, I}`. -/
-- Porting note: was @[nolint dangerous_instance]
instance isROrC_to_real : FiniteDimensional ℝ K :=
⟨⟨{1, I}, by
rw [eq_top_iff]
intro a _
rw [Finset.coe_insert, Finset.coe_singleton, Submodule.mem_span_insert]
refine' ⟨re a, im a • I, _, _⟩
· rw [Submodule.mem_span_singleton]
use im a
simp [re_add_im a, Algebra.smul_def, algebraMap_eq_ofReal]⟩⟩
⟨{1, I}, by simpa [Submodule.eq_top_iff', Submodule.mem_span_pair] using
fun x ↦ ⟨re x, im x, by simp [real_smul_eq_coe_mul]⟩⟩
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This proof relies a lot on the fact that simpa doesn't simplify the expression too much. I would prefer suffices _ by simpa [...]; exact _ or simp? [] says ...; exact

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@urkud I've changed it to use suffices, but for what it's worth, I disagree with:

This proof relies a lot on the fact that simpa doesn't simplify the expression too much.

The suffices I added was ∀ x : K, ∃ a b : ℝ, a • 1 + b • I = x (the goal after simp) which could only really be simplified further if we made IsROrC.real_smul_eq_coe_mul a simp lemma. But even if that did happen, this would be covered by the by simp [real_smul_eq_coe_mul]. Note also, making that a simp lemma would be a bit weird since it's inserting a coercion.

#align finite_dimensional.is_R_or_C_to_real FiniteDimensional.isROrC_to_real

variable (K E)
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