Si R es un anillo y a ∈ R, entonces a.0 = 0
Demostrar con Lean4 que si R es un anillo y a ∈ R, entonces
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a * 0 = 0 |
Para ello, completar la siguiente teoría de Lean4:
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import Mathlib.Algebra.Ring.Defs import Mathlib.Tactic variable {R : Type _} [Ring R] variable (a : R) example : a * 0 = 0 := sorry |
Demostración en lenguaje natural
Basta aplicar la propiedad cancelativa a
\[a.0 + a.0 = a.0 + 0\]
que se demuestra mediante la siguiente cadena de igualdades
\begin{align}
a.0 + a.0 &= a.(0 + 0) &&\text{[por la distributiva]} \\
&= a.0 &&\text{[por suma con cero]} \\
&= a.0 + 0 &&\text{[por suma con cero]}
\end{align}
Demostraciones con Lean4
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import Mathlib.Algebra.Ring.Defs import Mathlib.Tactic variable {R : Type _} [Ring R] variable (a : R) -- 1ª demostración -- =============== example : a * 0 = 0 := by have h : a * 0 + a * 0 = a * 0 + 0 := calc a * 0 + a * 0 = a * (0 + 0) := by rw [mul_add a 0 0] _ = a * 0 := by rw [add_zero 0] _ = a * 0 + 0 := by rw [add_zero (a * 0)] rw [add_left_cancel h] -- 2ª demostración -- =============== example : a * 0 = 0 := by have h : a * 0 + a * 0 = a * 0 + 0 := calc a * 0 + a * 0 = a * (0 + 0) := by rw [← mul_add] _ = a * 0 := by rw [add_zero] _ = a * 0 + 0 := by rw [add_zero] rw [add_left_cancel h] -- 3ª demostración -- =============== example : a * 0 = 0 := by have h : a * 0 + a * 0 = a * 0 + 0 := by rw [← mul_add, add_zero, add_zero] rw [add_left_cancel h] -- 4ª demostración -- =============== example : a * 0 = 0 := by have : a * 0 + a * 0 = a * 0 + 0 := calc a * 0 + a * 0 = a * (0 + 0) := by simp _ = a * 0 := by simp _ = a * 0 + 0 := by simp simp -- 5ª demostración -- =============== example : a * 0 = 0 := mul_zero a -- 6ª demostración -- =============== example : a * 0 = 0 := by simp |
Demostraciones interactivas
Se puede interactuar con las demostraciones anteriores en Lean 4 Web.
Referencias
- J. Avigad y P. Massot. Mathematics in Lean, p. 11.