Theorem: HurwitzZeta.hurwitzEvenFEPair_zero_symm
File: Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean

Proof steps:
state_before: ⊢ (hurwitzEvenFEPair 0).symm = hurwitzEvenFEPair 0
state_after: ⊢ {
      f :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.g,
      g :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.f,
      k :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.k,
      ε :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
              g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
              h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.ε⁻¹,
      f₀ :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.g₀,
      g₀ :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.f₀,
      hf_int := ⋯, hg_int := ⋯, hk := ⋯, hε := ⋯, h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ } =
    { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
      g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5, h_feq := ⋯,
      hf_top := ⋯, hg_top := ⋯ }
tactic: unfold hurwitzEvenFEPair WeakFEPair.symm
state_before: ⊢ {
      f :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.g,
      g :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.f,
      k :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.k,
      ε :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
              g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
              h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.ε⁻¹,
      f₀ :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.g₀,
      g₀ :=
        { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
            g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5,
            h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ }.f₀,
      hf_int := ⋯, hg_int := ⋯, hk := ⋯, hε := ⋯, h_feq := ⋯, hf_top := ⋯, hg_top := ⋯ } =
    { f := ofReal' ∘ evenKernel 0, g := ofReal' ∘ cosKernel 0, k := 1 / 2, ε := 1, f₀ := if 0 = 0 then 1 else 0,
      g₀ := 1, hf_int := ⋯, hg_int := ⋯, hk := hurwitzEvenFEPair.proof_4, hε := hurwitzEvenFEPair.proof_5, h_feq := ⋯,
      hf_top := ⋯, hg_top := ⋯ }
state_after: no goals
tactic: congr 1 <;> simp only [evenKernel_eq_cosKernel_of_zero, inv_one, if_true]