Nuprl Lemma : adjugate-property
∀[r:CRng]. ∀[n:ℕ]. ∀[M:Matrix(n;n;r)]. ((M*adj(M)) = |M|*I ∈ Matrix(n;n;r))
Proof
Definitions occuring in Statement :
adjugate: adj(M),
matrix-scalar-mul: k*M,
matrix-det: |M|,
identity-matrix: I,
matrix-times: (M*N),
matrix: Matrix(n;m;r),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
matrix: Matrix(n;m;r),
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
matrix-scalar-mul: k*M,
matrix-times: (M*N),
squash: ↓T,
crng: CRng,
rng: Rng,
so_lambda: λ2x y.t[x; y],
identity-matrix: I,
adjugate: adj(M),
so_apply: x[s1;s2],
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
matrix-ap: M[i,j],
le: A ≤ B,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
matrix-swap-rows: matrix-swap-rows(M;i;j),
determinant: determinant(n;r),
infix_ap: x f y,
isOdd: isOdd(n),
eq_int: (i =z j),
modulus: a mod n,
matrix-minor: matrix-minor(i;j;m),
mx: matrix(M[x; y]),
less_than: a < b,
subtract: n - m,
same-parity: same-parity(n;m),
ringeq_int_terms: t1 ≡ t2
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
mx_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_sig_wf,
matrix_ap_mx_lemma,
matrix_wf,
nat_wf,
crng_wf,
equal_wf,
matrix-det_wf,
matrix-ap_wf,
rng_times_one,
subtype_rel_self,
iff_weakening_equal,
rng_times_zero,
rng_wf,
upper_subtype_nat,
false_wf,
nequal-le-implies,
zero-add,
le_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
int_upper_properties,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
lelt_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
equal-rows-det,
not_wf,
matrix-swap-rows_wf,
exists_wf,
rng_times_wf,
isEven_wf,
rng_minus_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
matrix-minor_wf,
primrec-unroll,
lt_int_wf,
assert_of_lt_int,
intformeq_wf,
int_formula_prop_eq_lemma,
less_than_wf,
matrix-det-is-determinant,
rng_sum_wf,
subtract-add-cancel,
add-zero,
determinant_wf,
infix_ap_wf,
det-swap-rows,
equal-wf-base-T,
rng_minus_minus,
rng_minus_sum,
ge_wf,
int_seg_subtype_nat,
itermAdd_wf,
int_term_value_add_lemma,
iff_imp_equal_bool,
isOdd_wf,
btrue_wf,
assert_wf,
set_subtype_base,
top_wf,
add-member-int_seg2,
less_than_anti-reflexive,
equal-wf-base,
odd-implies,
even-implies,
bfalse_wf,
even-iff-not-odd,
isEven-add,
same-parity_wf,
odd-or-even,
assert_of_bor,
rng_times_over_minus,
itermMinus_wf,
itermMultiply_wf,
ringeq-iff-rsub-is-0,
ring_polynomial_null,
int-to-ring_wf,
ring_term_value_add_lemma,
ring_term_value_minus_lemma,
ring_term_value_mul_lemma,
ring_term_value_var_lemma,
ring_term_value_const_lemma,
int-to-ring-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
productElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
applyEquality,
imageElimination,
functionEquality,
imageMemberEquality,
baseClosed,
axiomEquality,
int_eqReduceTrueSq,
universeEquality,
int_eqReduceFalseSq,
hypothesis_subsumption,
lambdaFormation,
dependent_set_memberEquality,
equalityElimination,
promote_hyp,
productEquality,
addEquality,
applyLambdaEquality,
hyp_replacement,
intWeakElimination,
lessCases,
axiomSqEquality,
baseApply,
closedConclusion
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[M:Matrix(n;n;r)]. ((M*adj(M)) = |M|*I)
Date html generated:
2019_10_16-AM-11_28_16
Last ObjectModification:
2018_08_20-PM-09_47_22
Theory : matrices
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