/-- An orthonormal set in an `InnerProductSpace` is maximal, if and only if the orthogonal
complement of its span is empty. -/
theorem maximal_orthonormal_iff_orthogonalComplement_eq_bot (hv : Orthonormal 𝕜 ((↑) : v → E)) :
    (∀ (u) (_ : u ⊇ v), Orthonormal 𝕜 ((↑) : u → E) → u = v) ↔ (span 𝕜 v)ᗮ = ⊥ := by
  rw [Submodule.eq_bot_iff]
  constructor
  · contrapose!
    -- ** direction 1: nonempty orthogonal complement implies nonmaximal
    rintro ⟨x, hx', hx⟩
    -- take a nonzero vector and normalize it
    let e := (‖x‖⁻¹ : 𝕜) • x
    have he : ‖e‖ = 1 := by simp [norm_smul_inv_norm hx]
    have he' : e ∈ (span 𝕜 v)ᗮ := smul_mem' _ _ hx'
    have he'' : e ∉ v := by
      intro hev
      have : e = 0 := by
        have : e ∈ span 𝕜 v ⊓ (span 𝕜 v)ᗮ := ⟨subset_span hev, he'⟩
        simpa [(span 𝕜 v).inf_orthogonal_eq_bot] using this
      have : e ≠ 0 := hv.ne_zero ⟨e, hev⟩
      contradiction
    -- put this together with `v` to provide a candidate orthonormal basis for the whole space
    refine' ⟨insert e v, v.subset_insert e, ⟨_, _⟩, (ne_insert_of_not_mem v he'').symm⟩
    · -- show that the elements of `insert e v` have unit length
      rintro ⟨a, ha'⟩
      cases' eq_or_mem_of_mem_insert ha' with ha ha
      · simp [ha, he]
      · exact hv.1 ⟨a, ha⟩
    · -- show that the elements of `insert e v` are orthogonal
      have h_end : ∀ a ∈ v, ⟪a, e⟫ = 0 := by
        intro a ha
        exact he' a (Submodule.subset_span ha)
      rintro ⟨a, ha'⟩
      cases' eq_or_mem_of_mem_insert ha' with ha ha
      · rintro ⟨b, hb'⟩ hab'
        have hb : b ∈ v := by
          refine' mem_of_mem_insert_of_ne hb' _
          intro hbe'
          apply hab'
          simp [ha, hbe']
        rw [inner_eq_zero_symm]
        simpa [ha] using h_end b hb
      rintro ⟨b, hb'⟩ hab'
      cases' eq_or_mem_of_mem_insert hb' with hb hb
      · simpa [hb] using h_end a ha
      have : (⟨a, ha⟩ : v) ≠ ⟨b, hb⟩ := by
        intro hab''
        apply hab'
        simpa using hab''
      exact hv.2 this
  · -- ** direction 2: empty orthogonal complement implies maximal
    simp only [Subset.antisymm_iff]
    rintro h u (huv : v ⊆ u) hu
    refine' ⟨_, huv⟩
    intro x hxu
    refine' ((mt (h x)) (hu.ne_zero ⟨x, hxu⟩)).imp_symm _
    intro hxv y hy
    have hxv' : (⟨x, hxu⟩ : u) ∉ ((↑) ⁻¹' v : Set u) := by simp [huv, hxv]
    obtain ⟨l, hl, rfl⟩ :
      ∃ l ∈ Finsupp.supported 𝕜 𝕜 ((↑) ⁻¹' v : Set u), (Finsupp.total (↥u) E 𝕜 (↑)) l = y := by
      rw [← Finsupp.mem_span_image_iff_total]
      simp [huv, inter_eq_self_of_subset_left, hy]
    

Proof state:
case mpr.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
h : ∀ x ∈ (span 𝕜 v)ᗮ, x = 0
u : Set E
huv : v ⊆ u
hu : Orthonormal 𝕜 Subtype.val
x : E
hxu : x ∈ u
hxv : x ∉ v
hxv' : { val := x, property := hxu } ∉ Subtype.val ⁻¹' v
l : ↑u →₀ 𝕜
hl : l ∈ Finsupp.supported 𝕜 𝕜 (Subtype.val ⁻¹' v)
hy : (Finsupp.total (↑u) E 𝕜 Subtype.val) l ∈ span 𝕜 v
⊢ inner ((Finsupp.total (↑u) E 𝕜 Subtype.val) l) x = 0

Next tactic:
exact hu.inner_finsupp_eq_zero hxv' hl