Nuprl Lemma : det-fun+-at-identity
∀[n:ℕ]. ∀[J:ℕn + 1]. ∀[r:Rng]. ∀[d:det-fun(r;n + 1)].
((d matrix+(r;J;I)) = if isEven(J) then d I else -r (d I) fi ∈ |r|)
Proof
Definitions occuring in Statement :
matrix+: matrix+(r;j;M),
det-fun: det-fun(r;n),
identity-matrix: I,
isEven: isEven(n),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
apply: f a,
add: n + m,
natural_number: $n,
equal: s = t ∈ T,
rng: Rng,
rng_minus: -r,
rng_car: |r|
Definitions unfolded in proof :
true: True,
less_than': less_than'(a;b),
less_than: a < b,
nequal: a ≠ b ∈ T ,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
lelt: i ≤ j < k,
ge: i ≥ j ,
nat: ℕ,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
guard: {T},
implies: P ⇒ Q,
sq_type: SQType(T),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
int_seg: {i..j-},
rng: Rng,
prop: ℙ,
squash: ↓T,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
top: Top,
matrix-swap-rows: matrix-swap-rows(M;i;j),
matrix+: matrix+(r;j;M),
identity-matrix: I,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
le: A ≤ B,
subtype_rel: A ⊆r B,
det-fun: det-fun(r;n),
modulus: a mod n,
eq_int: (i =z j),
isEven: isEven(n),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
nat_wf,
rng_wf,
le_wf,
int_term_value_add_lemma,
itermAdd_wf,
decidable__le,
det-fun_wf,
top_wf,
rng_one_wf,
rng_zero_wf,
int_formula_prop_less_lemma,
intformless_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
intformnot_wf,
itermSubtract_wf,
subtract_wf,
less_than_wf,
assert_of_lt_int,
lt_int_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
rng_sig_wf,
rng_car_wf,
int_seg_wf,
true_wf,
squash_wf,
mx_wf,
matrix_ap_mx_lemma,
decidable__lt,
false_wf,
int_seg_subtype_nat,
ge_wf,
assert_wf,
btrue_wf,
isEven_wf,
iff_imp_equal_bool,
iff_weakening_equal,
matrix_wf,
rng_minus_wf,
identity-matrix_wf,
matrix+_wf,
equal-wf-base,
lelt_wf,
rng_minus_minus,
bfalse_wf,
not-even-succ-implies-even,
iff_wf,
assert_of_bnot,
not_wf,
bnot_wf,
subtract-add-cancel,
even-succ-implies-not-even
Rules used in proof :
axiomEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
imageMemberEquality,
sqequalAxiom,
lessCases,
int_eqReduceFalseSq,
promote_hyp,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
int_eqReduceTrueSq,
productElimination,
equalityElimination,
independent_functionElimination,
independent_isectElimination,
cumulativity,
instantiate,
unionElimination,
rename,
setElimination,
intEquality,
because_Cache,
natural_numberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
isectElimination,
imageElimination,
lambdaEquality,
applyEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
intWeakElimination,
functionExtensionality,
universeEquality,
applyLambdaEquality,
impliesFunctionality,
addLevel
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[J:\mBbbN{}n + 1]. \mforall{}[r:Rng]. \mforall{}[d:det-fun(r;n + 1)].
((d matrix+(r;J;I)) = if isEven(J) then d I else -r (d I) fi )
Date html generated:
2018_05_21-PM-09_37_41
Last ObjectModification:
2017_12_13-PM-06_25_26
Theory : matrices
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