English
Under surjectivity, comap is an embedding of submodules.
Русский
При сюръективности comap образует вложение подмодулей.
LaTeX
$$$\text{Injective}(\operatorname{comap})$ under surjectivity$$
Lean4
theorem ker_le_iff [RingHomSurjective τ₁₂] {p : Submodule R M} : ker f ≤ p ↔ ∃ y ∈ range f, f ⁻¹' { y } ⊆ p :=
by
constructor
· intro h
use 0
rw [← SetLike.mem_coe, coe_range]
exact ⟨⟨0, map_zero f⟩, h⟩
· rintro ⟨y, h₁, h₂⟩
rw [SetLike.le_def]
intro z hz
simp only [mem_ker] at hz
rw [← SetLike.mem_coe, coe_range, Set.mem_range] at h₁
obtain ⟨x, hx⟩ := h₁
have hx' : x ∈ p := h₂ hx
have hxz : z + x ∈ p := by
apply h₂
simp [hx, hz]
suffices z + x - x ∈ p by simpa only [this, add_sub_cancel_right]
exact p.sub_mem hxz hx'
