Nuprl Lemma : det-fun-is-determinant
∀[r:CRng]. ∀[n:ℕ]. ∀[d:det-fun(r;n)]. (d = (λM.((d I) * (determinant(n;r) M))) ∈ (Matrix(n;n;r) ⟶ |r|))
Proof
Definitions occuring in Statement :
determinant: determinant(n;r),
det-fun: det-fun(r;n),
identity-matrix: I,
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
crng: CRng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
crng: CRng,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
matrix: Matrix(n;m;r),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
rng: Rng,
squash: ↓T,
det-fun: det-fun(r;n),
determinant: determinant(n;r),
decidable: Dec(P),
or: P ∨ Q,
subtract: n - m,
sq_type: SQType(T),
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
nequal: a ≠ b ∈ T ,
so_apply: x[s1;s2],
matrix-mul-row: matrix-mul-row(r;k;i;M),
matrix-ap: M[i,j],
mx: matrix(M[x; y]),
less_than: a < b,
label: ...$L... t,
rng_zero: 0,
pi1: fst(t),
pi2: snd(t),
matrix+: matrix+(r;j;M),
matrix-minor: matrix-minor(i;j;m),
row-op: row-op(r;a;b;k;M),
ringeq_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
det-fun_wf,
le_wf,
subtract-1-ge-0,
nat_wf,
crng_wf,
false_wf,
matrix_wf,
int_seg_wf,
int_seg_properties,
iff_weakening_equal,
subtype_rel_self,
identity-matrix_wf,
rng_times_one,
rng_car_wf,
true_wf,
squash_wf,
equal_wf,
primrec0_lemma,
det-fun+,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
subtract-add-cancel,
decidable__lt,
subtype_base_sq,
int_subtype_base,
add-associates,
add-swap,
add-commutes,
zero-add,
rng_times_wf,
det-fun+-at-identity,
determinant_wf,
matrix-minor_wf,
istype-false,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
matrix-ap_wf,
istype-universe,
rng_sum_wf,
rng_wf,
mx_wf,
eq_int_wf,
assert_of_eq_int,
rng_one_wf,
neg_assert_of_eq_int,
rng_zero_wf,
rng_sig_wf,
set_subtype_base,
lelt_wf,
rng_times_zero,
rng_plus_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-top,
rng_sum_unroll_base,
rng_plus_comm,
rng_plus_zero,
infix_ap_wf,
rng_sum_unroll_hi,
rng_plus_assoc,
rng_plus_ac_1,
matrix_ap_mx_lemma,
decidable__equal_int,
det-fun-zero-row,
det-fun-row-op,
rng_minus_wf,
less_than_anti-reflexive,
rng_times_over_minus,
rng_plus_inv,
rng_minus_zero,
rng_times_sum_l,
isEven_wf,
crng_times_ac_1,
itermAdd_wf,
itermMultiply_wf,
itermMinus_wf,
ringeq-iff-rsub-is-0,
ring_polynomial_null,
int-to-ring_wf,
ring_term_value_add_lemma,
ring_term_value_mul_lemma,
ring_term_value_var_lemma,
ring_term_value_minus_lemma,
ring_term_value_const_lemma,
int-to-ring-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
dependent_set_memberEquality_alt,
because_Cache,
lambdaFormation,
dependent_set_memberEquality,
intEquality,
dependent_pairFormation,
instantiate,
baseClosed,
imageMemberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
productElimination,
voidEquality,
isect_memberEquality,
functionExtensionality,
isect_memberFormation,
unionElimination,
productIsType,
addEquality,
cumulativity,
applyLambdaEquality,
equalityElimination,
equalityIsType2,
baseApply,
closedConclusion,
promote_hyp,
equalityIsType1,
functionIsType,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
lessCases,
axiomSqEquality,
hyp_replacement,
equalityIsType4
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[d:det-fun(r;n)]. (d = (\mlambda{}M.((d I) * (determinant(n;r) M))))
Date html generated:
2019_10_16-AM-11_27_58
Last ObjectModification:
2018_10_10-PM-03_06_14
Theory : matrices
Home
Index