Nuprl Lemma : det-diagonal
∀[r:CRng]. ∀[n:ℕ]. ∀[F:ℕn ⟶ |r|]. (|diagonal-matrix(r;i.F[i])| = (Π(r) 0 ≤ i < n. F[i]) ∈ |r|)
Proof
Definitions occuring in Statement :
diagonal-matrix: diagonal-matrix(r;x.F[x]),
matrix-det: |M|,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
rng_prod: rng_prod,
crng: CRng,
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,
rng: Rng,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
diagonal-matrix: diagonal-matrix(r;x.F[x]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
infix_ap: x f y,
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
ringeq_int_terms: t1 ≡ t2,
matrix: Matrix(n;m;r),
matrix-minor: matrix-minor(i;j;m),
mx: matrix(M[x; y]),
less_than: a < b,
less_than': less_than'(a;b),
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
int_seg_wf,
rng_car_wf,
equal_wf,
squash_wf,
true_wf,
matrix-det-dim0,
rng_prod_wf,
subtype_rel_self,
iff_weakening_equal,
rng_prod_unroll_base,
rng_one_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
crng_wf,
expand-det-by-row,
le_wf,
decidable__lt,
lelt_wf,
diagonal-matrix_wf,
rng_sum_unroll_hi,
infix_ap_wf,
rng_times_wf,
isEven_wf,
bool_wf,
eqtt_to_assert,
matrix_ap_mx_lemma,
rng_zero_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
rng_minus_wf,
matrix-det_wf,
matrix-minor_wf,
rng_plus_wf,
rng_sum_is_0,
eq_int_wf,
assert_of_eq_int,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
itermAdd_wf,
itermMultiply_wf,
itermMinus_wf,
ringeq-iff-rsub-is-0,
mx_wf,
ring_polynomial_null,
int-to-ring_wf,
ring_term_value_add_lemma,
ring_term_value_mul_lemma,
ring_term_value_const_lemma,
int-to-ring-zero,
ring_term_value_var_lemma,
ring_term_value_minus_lemma,
two-mul,
btrue_wf,
assert-isEven,
equal-wf-base,
int_subtype_base,
matrix_wf,
rng_wf,
lt_int_wf,
assert_of_lt_int,
top_wf,
rng_prod_unroll_hi
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
functionEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
unionElimination,
dependent_set_memberEquality,
functionExtensionality,
addEquality,
equalityElimination,
promote_hyp,
cumulativity,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
multiplyEquality,
baseApply,
closedConclusion,
hyp_replacement,
applyLambdaEquality,
lessCases,
sqequalAxiom
Latex:
\mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}]. \mforall{}[F:\mBbbN{}n {}\mrightarrow{} |r|]. (|diagonal-matrix(r;i.F[i])| = (\mPi{}(r) 0 \mleq{} i < n. F[i]))
Date html generated:
2018_05_21-PM-09_39_32
Last ObjectModification:
2018_05_19-PM-04_31_08
Theory : matrices
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