Nuprl Lemma : expand-det-by-column
∀[n:ℕ]. ∀[j:ℕn]. ∀[r:CRng]. ∀[M:Matrix(n;n;r)].
(|M| = (Σ(r) 0 ≤ i < n. if isEven(i + j) then M[i,j] else -r M[i,j] fi * |matrix-minor(i;j;M)|) ∈ |r|)
Proof
Definitions occuring in Statement :
matrix-minor: matrix-minor(i;j;m),
matrix-det: |M|,
matrix-ap: M[i,j],
matrix: Matrix(n;m;r),
isEven: isEven(n),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
apply: f a,
subtract: n - m,
add: n + m,
natural_number: $n,
equal: s = t ∈ T,
rng_sum: rng_sum,
crng: CRng,
rng_times: *,
rng_minus: -r,
rng_car: |r|
Definitions unfolded in proof :
so_apply: x[s],
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
so_lambda: λ2x.t[x],
matrix-times: (M*N),
adjugate: adj(M),
int_seg: {i..j-},
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
all: ∀x:A. B[x],
matrix-scalar-mul: k*M,
identity-matrix: I,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
rng: Rng,
crng: CRng,
nat: ℕ,
prop: ℙ,
squash: ↓T,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_minus_wf,
nat_wf,
crng_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
rng_times_wf,
infix_ap_wf,
matrix-minor_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
subtract_wf,
crng_times_comm,
eqtt_to_assert,
bool_wf,
isEven_wf,
add-commutes,
rng_wf,
rng_sum_wf,
rng_times_one,
rng_car_wf,
matrix_ap_mx_lemma,
iff_weakening_equal,
identity-matrix_wf,
matrix-det_wf,
matrix-scalar-mul_wf,
adjugate-property2,
equal_wf,
rng_sig_wf,
int_seg_wf,
matrix_wf,
true_wf,
squash_wf,
matrix-ap_wf,
rng_times_over_minus
Rules used in proof :
axiomEquality,
cumulativity,
instantiate,
promote_hyp,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
dependent_set_memberEquality,
equalityElimination,
unionElimination,
lambdaFormation,
addEquality,
functionEquality,
int_eqReduceTrueSq,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
universeEquality,
because_Cache,
rename,
setElimination,
intEquality,
natural_numberEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[j:\mBbbN{}n]. \mforall{}[r:CRng]. \mforall{}[M:Matrix(n;n;r)].
(|M| = (\mSigma{}(r) 0 \mleq{} i < n. if isEven(i + j) then M[i,j] else -r M[i,j] fi * |matrix-minor(i;j;M)|))
Date html generated:
2018_05_21-PM-09_39_22
Last ObjectModification:
2017_12_14-PM-03_54_58
Theory : matrices
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