Nuprl Lemma : adjugate-property2
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ((adj(M)*M) = |M|*I ∈ Matrix(n;n;r))
Proof
Definitions occuring in Statement :
adjugate: adj(M),
matrix-scalar-mul: k*M,
matrix-det: |M|,
identity-matrix: I,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
top: Top,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
rng: Rng,
crng: CRng,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q
Lemmas referenced :
crng_wf,
nat_wf,
matrix_wf,
iff_weakening_equal,
identity-matrix_wf,
matrix-det_wf,
matrix-scalar-mul_wf,
adjugate_wf,
matrix-times_wf,
true_wf,
squash_wf,
equal_wf,
matrix-transpose-twice,
matrix-transpose_wf,
matrix-transpose-times,
rng_wf,
adjugate-transpose,
transpose-scalar-mul,
adjugate-property,
rng_car_wf,
rng_sig_wf,
det-transpose,
transpose-identity-matrix
Rules used in proof :
voidEquality,
voidElimination,
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
applyLambdaEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
intEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. ((adj(M)*M) = |M|*I)
Date html generated:
2018_05_21-PM-09_39_15
Last ObjectModification:
2017_12_14-PM-00_51_55
Theory : matrices
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