Theorems · Theorem · linear algebra
Module.Basis.ext_elem
∀ {ι : Type u_10} {R : Type u_11} {M : Type u_12} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(b : Module.Basis ι R M) {x y : M}, (∀ (i : ι), (b.repr x) i = (b.repr y) i) → x = yAlias of the reverse direction of Module.Basis.ext_elem_iff.
Two elements are equal iff their coordinates are equal.
- Defined in
- Mathlib.LinearAlgebra.Basis.Defs
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Finsuppstatement · cited by 5,255
- LinearEquivstatement · cited by 3,317
- Module.Basisstatement and proof · cited by 1,477
- Module.Basis.reprstatement · cited by 498
- Module.Basis.ext_elem_iffproof · cited by 10
Cited by11
Results whose statement or proof uses this declaration.
- Algebra.norm_eq_zero_iffproof · cited by 6
- Module.Basis.eval_injectiveproof · cited by 2
- Basis.piTensorProduct_applyproof · cited by 2
- ZSpan.fract_fractproof · cited by 1
- eq_bot_of_generator_maximal_submoduleImage_eq_zeroproof · cited by 1
- ZSpan.ceil_eq_self_of_memproof · cited by 0
- ZSpan.coe_floor_selfproof · cited by 0
- ZSpan.coe_fract_selfproof · cited by 0
- ZSpan.floor_eq_self_of_memproof · cited by 0
- eq_bot_of_generator_maximal_map_eq_zeroproof · cited by 0
- PiTensorProduct.dualDistrib_dualDistribInvOfBasis_right_inverseproof · cited by 0