Nuprl Lemma : det-fun-zero-row
∀[r:Rng]. ∀[n:ℕ]. ∀[d:det-fun(r;n)]. ∀[M:Matrix(n;n;r)]. (d M) = 0 ∈ |r| supposing ∃i:ℕn. ∀j:ℕn. (M[i,j] = 0 ∈ |r|)
Proof
Definitions occuring in Statement :
det-fun: det-fun(r;n),
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
natural_number: $n,
equal: s = t ∈ T,
rng: Rng,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
nat: ℕ,
squash: ↓T,
prop: ℙ,
rng: Rng,
all: ∀x:A. B[x],
and: P ∧ Q,
det-fun: det-fun(r;n),
exists: ∃x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
int_seg: {i..j-},
mx: matrix(M[x; y]),
matrix-ap: M[i,j],
matrix-mul-row: matrix-mul-row(r;k;i;M),
matrix: Matrix(n;m;r),
infix_ap: x f y,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rng_wf,
nat_wf,
det-fun_wf,
matrix-ap_wf,
all_wf,
int_seg_wf,
exists_wf,
matrix_wf,
rng_times_zero,
rng_car_wf,
true_wf,
squash_wf,
equal_wf,
rng_zero_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
rng_times_wf,
int_subtype_base,
iff_weakening_equal
Rules used in proof :
axiomEquality,
isect_memberEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
functionExtensionality,
because_Cache,
universeEquality,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
sqequalRule,
equalitySymmetry,
hyp_replacement,
isectElimination,
extract_by_obid,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
rename,
setElimination,
thin,
productElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
int_eqReduceFalseSq,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation,
int_eqReduceTrueSq,
independent_isectElimination,
equalityElimination,
unionElimination,
lambdaFormation,
intEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[n:\mBbbN{}]. \mforall{}[d:det-fun(r;n)]. \mforall{}[M:Matrix(n;n;r)].
(d M) = 0 supposing \mexists{}i:\mBbbN{}n. \mforall{}j:\mBbbN{}n. (M[i,j] = 0)
Date html generated:
2018_05_21-PM-09_36_49
Last ObjectModification:
2017_12_13-PM-07_07_03
Theory : matrices
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