Nuprl Lemma : null-space-unique
∀[r:IntegDom{i}]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)].
  ∀[u:Column(n;r)]. (((M*u) = 0 ∈ Column(n;r)) 
⇒ (u = 0 ∈ Column(n;r))) supposing ¬(|M| = 0 ∈ |r|)
Proof
Definitions occuring in Statement : 
matrix-det: |M|
, 
zero-matrix: 0
, 
matrix-times: (M*N)
, 
matrix: Matrix(n;m;r)
, 
nat: ℕ
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
natural_number: $n
, 
equal: s = t ∈ T
, 
integ_dom: IntegDom{i}
, 
rng_zero: 0
, 
rng_car: |r|
Definitions unfolded in proof : 
true: True
, 
not: ¬A
, 
false: False
, 
less_than': less_than'(a;b)
, 
and: P ∧ Q
, 
le: A ≤ B
, 
prop: ℙ
, 
rng: Rng
, 
crng: CRng
, 
nat: ℕ
, 
implies: P 
⇒ Q
, 
uimplies: b supposing a
, 
integ_dom: IntegDom{i}
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
so_apply: x[s1;s2]
, 
so_lambda: λ2x y.t[x; y]
, 
top: Top
, 
all: ∀x:A. B[x]
, 
zero-matrix: 0
, 
matrix-ap: M[i,j]
, 
matrix: Matrix(n;m;r)
, 
matrix-scalar-mul: k*M
, 
integ_dom_p: IsIntegDom(r)
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
le_wf, 
false_wf, 
integ_dom_wf, 
nat_wf, 
rng_zero_wf, 
matrix-det_wf, 
rng_car_wf, 
not_wf, 
zero-matrix_wf, 
matrix_wf, 
equal_wf, 
identity-matrix_wf, 
adjugate_wf, 
matrix-times_wf, 
adjugate-property2, 
matrix-times-0-right, 
squash_wf, 
true_wf, 
matrix-times-assoc, 
rng_wf, 
matrix-scalar-mul-times, 
matrix-scalar-mul_wf, 
rng_sig_wf, 
matrix-times-id-left, 
iff_weakening_equal, 
int_seg_wf, 
matrix_ap_mx_lemma, 
matrix-ap_wf, 
crng_times_comm
Rules used in proof : 
equalitySymmetry, 
equalityTransitivity, 
independent_pairFormation, 
dependent_set_memberEquality, 
isect_memberEquality, 
axiomEquality, 
dependent_functionElimination, 
lambdaEquality, 
sqequalRule, 
natural_numberEquality, 
because_Cache, 
applyLambdaEquality, 
lambdaFormation, 
hypothesisEquality, 
rename, 
setElimination, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
hypothesis, 
isect_memberFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
extract_by_obid, 
introduction, 
cut, 
applyEquality, 
imageElimination, 
universeEquality, 
intEquality, 
imageMemberEquality, 
baseClosed, 
independent_isectElimination, 
productElimination, 
independent_functionElimination, 
voidEquality, 
voidElimination, 
functionExtensionality, 
hyp_replacement
Latex:
\mforall{}[r:IntegDom\{i\}].  \mforall{}[n:\mBbbN{}].  \mforall{}[M:Matrix(n;n;r)].
    \mforall{}[u:Column(n;r)].  (((M*u)  =  0)  {}\mRightarrow{}  (u  =  0))  supposing  \mneg{}(|M|  =  0)
Date html generated:
2018_05_21-PM-09_39_19
Last ObjectModification:
2017_12_20-PM-06_11_20
Theory : matrices
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