`ext`

proved

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module: IndisputableMonolith.Algebra.CostAlgebra
domain: Algebra
line: 783 · github
papers citing: none yet

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IndisputableMonolith.Algebra.CostAlgebra on GitHub at line 783.

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depends on

formal source

 780    J (f x).1 = J x.1 :=
 781  f.2.2 x
 782
 783@[ext] theorem ext {f g : JAut} (h : ∀ x : PosReal, f x = g x) : f = g := by
 784  apply Subtype.ext
 785  funext x
 786  exact h x
 787
 788/-- The identity automorphism. -/
 789def id : JAut :=
 790  ⟨fun x => x, by
 791    constructor
 792    · intro _ _
 793      rfl
 794    · intro _
 795      rfl⟩
 796
 797/-- The reciprocal automorphism. -/
 798noncomputable def reciprocal : JAut :=
 799  ⟨posInv, by
 800    constructor
 801    · intro x y
 802      apply Subtype.ext
 803      simp [posMul, posInv, mul_comm]
 804    · intro x
 805      simpa [posInv] using (J_reciprocal x.1 x.2).symm⟩
 806
 807/-- Composition of `J`-automorphisms. -/
 808def comp (g f : JAut) : JAut :=
 809  ⟨fun x => g (f x), by
 810    constructor
 811    · intro x y
 812      change g (f (posMul x y)) = posMul (g (f x)) (g (f y))
 813      rw [f.multiplicative x y, g.multiplicative (f x) (f y)]

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