Nuprl Lemma : invertible-matrix-iff-det
∀r:CRng. ∀n:ℕ. ∀A:Matrix(n;n;r). (invertible-matrix(r;n;A) ⇐⇒ |A| | 1 in r)
Proof
Definitions occuring in Statement :
invertible-matrix: invertible-matrix(r;n;A),
matrix-det: |M|,
matrix: Matrix(n;m;r),
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
ring_divs: a | b in r,
crng: CRng,
rng_one: 1
Definitions unfolded in proof :
nat: ℕ,
rng: Rng,
rev_implies: P ⇐ Q,
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],
exists: ∃x:A. B[x],
invertible-matrix: invertible-matrix(r;n;A),
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
infix_ap: x f y,
ring_divs: a | b in r
Lemmas referenced :
crng_wf,
nat_wf,
matrix_wf,
rng_one_wf,
matrix-det_wf,
ring_divs_wf,
invertible-matrix_wf,
adjugate-property,
iff_weakening_equal,
rng_wf,
rng_car_wf,
true_wf,
squash_wf,
equal_wf,
det-times,
det-id,
rng_times_wf,
crng_times_comm,
identity-matrix_wf,
matrix-times_wf,
adjugate_wf,
matrix-scalar-mul_wf,
matrix-times-scalar-mul,
rng_sig_wf,
matrix-scalar-mul-mul,
matrix-scalar-mul-1
Rules used in proof :
because_Cache,
rename,
setElimination,
independent_pairFormation,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
productElimination,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
dependent_pairFormation,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}r:CRng. \mforall{}n:\mBbbN{}. \mforall{}A:Matrix(n;n;r). (invertible-matrix(r;n;A) \mLeftarrow{}{}\mRightarrow{} |A| | 1 in r)
Date html generated:
2018_05_21-PM-09_39_59
Last ObjectModification:
2017_12_14-PM-01_43_02
Theory : matrices
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