Nuprl Lemma : Cramers-rule
∀[r:CRng]. ∀[n:ℕ]. ∀[A:Matrix(n;n;r)]. ∀[c:{c:|r|| (c * |A|) = 1 ∈ |r|} ]. ∀[b:Column(n;r)].
((A*c*matrix(|matrix(if y=j then b[x,0] else A[x,y])|)) = b ∈ Column(n;r))
Proof
Definitions occuring in Statement :
matrix-scalar-mul: k*M,
matrix-det: |M|,
matrix-times: (M*N),
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
int_eq: if a=b then c else d,
set: {x:A| B[x]} ,
natural_number: $n,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
true: True,
so_apply: x[s],
infix_ap: x f y,
so_lambda: λ2x.t[x],
rng: Rng,
crng: CRng,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
crng_wf,
nat_wf,
rng_one_wf,
matrix-det_wf,
rng_times_wf,
equal_wf,
rng_car_wf,
set_wf,
matrix_wf,
le_wf,
false_wf,
adj-solution-property,
squash_wf,
true_wf,
matrix-times_wf,
rng_wf,
adj-solution-column,
iff_weakening_equal
Rules used in proof :
applyEquality,
lambdaEquality,
because_Cache,
axiomEquality,
isect_memberEquality,
independent_isectElimination,
hypothesis,
lambdaFormation,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
intEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[A:Matrix(n;n;r)]. \mforall{}[c:\{c:|r|| (c * |A|) = 1\} ]. \mforall{}[b:Column(n;r)].
((A*c*matrix(|matrix(if y=j then b[x,0] else A[x,y])|)) = b)
Date html generated:
2018_05_21-PM-09_40_26
Last ObjectModification:
2017_12_14-PM-04_18_11
Theory : matrices
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