Nuprl Lemma : invertible-matrix-iff-left
∀r:CRng. ∀n:ℕ. ∀A:Matrix(n;n;r). (invertible-matrix(r;n;A) ⇐⇒ ∃B:Matrix(n;n;r). ((B*A) = I ∈ Matrix(n;n;r)))
Proof
Definitions occuring in Statement :
invertible-matrix: invertible-matrix(r;n;A),
identity-matrix: I,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
rev_implies: P ⇐ Q,
rng: Rng,
crng: CRng,
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
ring_divs: a | b in r,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
infix_ap: x f y
Lemmas referenced :
crng_wf,
nat_wf,
iff_wf,
invertible-matrix_wf,
invertible-matrix-iff-det,
identity-matrix_wf,
matrix-times_wf,
equal_wf,
matrix_wf,
exists_wf,
rng_one_wf,
matrix-det_wf,
ring_divs_wf,
adjugate-property2,
adjugate_wf,
matrix-scalar-mul_wf,
squash_wf,
true_wf,
matrix-scalar-mul-times,
rng_car_wf,
rng_sig_wf,
iff_weakening_equal,
matrix-scalar-mul-mul,
matrix-scalar-mul-1,
rng_times_wf,
rng_wf,
det-times,
det-id
Rules used in proof :
independent_functionElimination,
dependent_functionElimination,
impliesFunctionality,
productElimination,
addLevel,
lambdaEquality,
sqequalRule,
because_Cache,
hypothesis,
rename,
setElimination,
independent_pairFormation,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_pairFormation,
natural_numberEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}r:CRng. \mforall{}n:\mBbbN{}. \mforall{}A:Matrix(n;n;r). (invertible-matrix(r;n;A) \mLeftarrow{}{}\mRightarrow{} \mexists{}B:Matrix(n;n;r). ((B*A) = I))
Date html generated:
2018_05_21-PM-09_40_02
Last ObjectModification:
2017_12_14-PM-01_49_13
Theory : matrices
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