primeSpectrumProd_symm_inl_asIdeal

English

Under the bijection PrimeSpectrum(R) ⊕ PrimeSpectrum(S) ≃ PrimeSpectrum(R × S), the preimage of Sum.inl x corresponds to the ideal x.asIdeal × ⊤.

Русский

Под биекцией PrimeSpectrum(R) ⊕ PrimeSpectrum(S) ≃ PrimeSpectrum(R × S) образующейSum.inl x соответствует идеал x.asIdeal × ⊤.

LaTeX

$$$$\\\\bigl((\\\\text{primeSpectrumProd}(R,S))\\\\text{ symm} \\\\circ \\\\text{Sum.inl} x\\\\bigr).asIdeal = x.asIdeal \\\\top$$$$

Lean4

@[simp]
theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
    ((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ :=
  by
  cases x
  rfl

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