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[Merged by Bors] - feat(Algebra/DirectSum/Decomposition): degree of a non-zero homogeneous element is unique #9597

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[Merged by Bors] - feat(Algebra/DirectSum/Decomposition): degree of a non-zero homogeneous element is unique #9597

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jjaassoonn Jan 9, 2024
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更新 Decomposition.lean
jjaassoonn Jan 10, 2024
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4 changes: 4 additions & 0 deletions Mathlib/Algebra/DirectSum/Decomposition.lean
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Expand Up @@ -143,6 +143,10 @@ theorem decompose_of_mem_ne {x : M} {i j : ι} (hx : x ∈ ℳ i) (hij : i ≠ j
rw [decompose_of_mem _ hx, DirectSum.of_eq_of_ne _ _ _ _ hij, ZeroMemClass.coe_zero]
#align direct_sum.decompose_of_mem_ne DirectSum.decompose_of_mem_ne

theorem degree_eq_of_mem_mem {x : M} {i j : ι} (hxi : x ∈ ℳ i) (hxj : x ∈ ℳ j) (hx : x ≠ 0) :
i = j := by
contrapose! hx; rw [← decompose_of_mem_same ℳ hxj, decompose_of_mem_ne ℳ hxi hx]

/-- If `M` is graded by `ι` with degree `i` component `ℳ i`, then it is isomorphic as
an additive monoid to a direct sum of components. -/
-- Porting note: deleted [simps] and added the corresponding lemmas by hand
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