Nuprl Lemma : adj-solution-property
∀[r:CRng]. ∀[n,m:ℕ]. ∀[A:Matrix(n;n;r)]. ∀[c:|r|]. ∀[b:Matrix(n;m;r)].
(A*adj-solution(r;n;A;c;b)) = b ∈ Matrix(n;m;r) supposing (c * |A|) = 1 ∈ |r|
Proof
Definitions occuring in Statement :
adj-solution: adj-solution(r;n;A;c;b),
matrix-det: |M|,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
true: True,
nat: ℕ,
infix_ap: x f y,
rng: Rng,
crng: CRng,
prop: ℙ,
adj-solution: adj-solution(r;n;A;c;b),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
adjugate_wf,
matrix-times_wf,
crng_wf,
nat_wf,
matrix_wf,
rng_one_wf,
matrix-det_wf,
rng_times_wf,
rng_car_wf,
equal_wf,
squash_wf,
true_wf,
matrix-times-scalar-mul,
matrix-scalar-mul_wf,
rng_sig_wf,
matrix-times-assoc,
rng_wf,
adjugate-property,
iff_weakening_equal,
matrix-scalar-mul-times,
identity-matrix_wf,
matrix-scalar-mul-mul,
matrix-scalar-mul-1,
matrix-times-id-left
Rules used in proof :
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
isect_memberEquality,
sqequalRule,
because_Cache,
applyEquality,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaEquality,
imageElimination,
universeEquality,
intEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[A:Matrix(n;n;r)]. \mforall{}[c:|r|]. \mforall{}[b:Matrix(n;m;r)].
(A*adj-solution(r;n;A;c;b)) = b supposing (c * |A|) = 1
Date html generated:
2018_05_21-PM-09_40_16
Last ObjectModification:
2017_12_14-PM-03_37_21
Theory : matrices
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