rangeKerLift_injective — Mathlib

English

The map rangeKerLift φ is surjective onto φ.range.

Русский

Карта rangeKerLift φ сюръективна на диапазон φ.

LaTeX

$$Surjective (rangeKerLift φ)$$

Lean4

@[to_additive]
theorem rangeKerLift_injective : Injective (rangeKerLift φ) := fun a b =>
  Quotient.inductionOn₂' a b fun a b (h : φ.rangeRestrict a = φ.rangeRestrict b) =>
    Quotient.sound' <| by
      rw [leftRel_apply, ← ker_rangeRestrict, mem_ker, φ.rangeRestrict.map_mul, ← h, φ.rangeRestrict.map_inv,
        inv_mul_cancel]

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